#481774

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#35144	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

938 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cg,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgh,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318439, #72318727, #72318757
maximal normal subgroup of	#72318439, #72318727, #72318757

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388970);