#6378181

order	512
nilpotency class	3
isoclinism	isoclinic class of #60771
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60915	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cfi,d^b=di,g^c=gi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308458, #72308699, #72308757
maximal normal subgroup of	#72308458, #72308699, #72308757

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6285377);