#368861

order	512
nilpotency class	3
isoclinism	isoclinic class of #368859
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14023
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1634 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)(3,15,18,9,19,32,28,33)(4,20,22,37,23,36,29,10)(5,24,25,11,27,38,26,39)(16,30,44,40,45,34,43,49)(17,31,47,56,48,57,46,41)(21,51,53,59,54,35,52,50)(42,61,62,55,64,60,63,58);(1,3,8,18,7,19,6,28)(2,9,12,32,14,33,13,15)(4,16,22,44,23,45,29,43)(5,17,25,47,27,48,26,46)(10,34,20,49,37,30,36,40)(11,31,38,56,39,57,24,41)(21,42,53,62,54,64,52,63)(35,58,50,61,51,55,59,60);(1,4,8,29,7,23,6,22)(2,10,13,20,14,37,12,36)(3,16,18,43,19,45,28,44)(5,21,25,52,27,54,26,53)(9,30,15,40,33,34,32,49)(11,35,24,50,39,51,38,59)(17,42,47,63,48,64,46,62)(31,55,41,60,57,58,56,61);(1,5)(2,11)(3,17)(4,21)(6,26)(7,27)(8,25)(9,31)(10,35)(12,38)(13,24)(14,39)(15,41)(16,42)(18,47)(19,48)(20,50)(22,53)(23,54)(28,46)(29,52)(30,55)(32,56)(33,57)(34,58)(36,59)(37,51)(40,60)(43,63)(44,62)(45,64)(49,61)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,e^2=i,f^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^b=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319703, #72319962, #72320069
maximal normal subgroup of	#72319703, #72319962, #72320069

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276057);