#360762

order	512
nilpotency class	3
isoclinism	isoclinic class of #25798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13880
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1299 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319834, #72319855, #72319897, …
maximal normal subgroup of	#72319834, #72319855, #72319897, …
maximal quotient of	#72343144

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267958);