#392428

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10,15,19,14,17,13,29)(11,20,24,30,23,33,22,16)(12,25,28,40,27,18,26,32)(21,35,38,31,37,34,36,39);(1,3,4,7)(2,5,6,8)(9,11,15,24,14,23,13,22)(10,16,29,33,17,30,19,20)(12,21,28,38,27,37,26,36)(18,31,40,35,25,39,32,34);(2,6)(5,8)(10,17)(16,30)(18,25)(19,29)(20,33)(31,39)(32,40)(34,35);(9,12,13,26,14,27,15,28)(10,18,19,32,17,25,29,40)(11,21,22,36,23,37,24,38)(16,31,20,34,30,39,33,35)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^d=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306938, #72306946, #72306950, …
maximal normal subgroup of	#72306938, #72306946, #72306950, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299624);