#374102

order	512
nilpotency class	3
isoclinism	isoclinic class of #24408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14134
prime radical	#60366	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2380 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342962, #72342963, #72342979
maximal normal subgroup of	#72342962, #72342963, #72342979
maximal quotient of	#72343062

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281298);