#454674

order	512
nilpotency class	3
isoclinism	isoclinic class of #454673
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16255
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1752 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,8,12,7,14,6,13)(3,15,18,9,19,33,27,32)(4,10,28,36,23,38,22,37)(5,11,25,42,26,41,29,40)(16,44,20,30,46,34,51,59)(17,47,52,61,50,43,49,31)(21,35,24,56,54,62,55,39)(45,60,53,63,57,64,48,58);(1,3)(2,9)(4,20)(5,17)(6,27)(7,19)(8,18)(10,34)(11,31)(12,33)(13,15)(14,32)(16,22)(21,53)(23,51)(24,57)(25,52)(26,50)(28,46)(29,49)(30,37)(35,60)(36,59)(38,44)(39,58)(40,43)(41,61)(42,47)(45,55)(48,54)(56,63)(62,64);(1,4,7,23)(2,10,14,38)(3,16,19,46)(5,24,26,55)(6,28,8,22)(9,30,32,59)(11,39,41,56)(12,37,13,36)(15,34,33,44)(17,48,50,53)(18,51,27,20)(21,29,54,25)(31,60,61,64)(35,42,62,40)(43,63,47,58)(45,52,57,49);(1,5)(2,11)(3,17)(4,21)(6,29)(7,26)(8,25)(9,31)(10,35)(12,42)(13,40)(14,41)(15,43)(16,45)(18,52)(19,50)(20,53)(22,55)(23,54)(24,28)(27,49)(30,58)(32,61)(33,47)(34,60)(36,56)(37,39)(38,62)(44,64)(46,57)(48,51)(59,63)

Polycyclic presentation:

2^9|a^2=ef,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264941, #72264949, #72264996
maximal normal subgroup of	#72264941, #72264949, #72264996
maximal quotient of	#72265619

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361870);