#482640

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊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×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

926 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)(3,4)(7,8)(9,10)(11,19,20,32)(12,23,25,17)(13,26,28,18)(14,16)(15,22)(21,35)(24,31)(27,30)(29,36,37,39)(33,34)(38,40);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,21)(13,27)(14,20)(16,30)(17,29)(18,32)(19,26)(22,28)(23,37)(24,39)(25,34)(31,35)(33,38)(36,40);(1,3,5,4)(2,6,7,8)(10,16)(11,20)(17,23)(18,26)(21,34)(22,27)(31,38)(36,39);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,21)(13,24)(14,25)(15,29)(16,23)(18,31)(19,33)(20,34)(22,36)(26,38)(27,39)(28,40)(30,37)(32,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318439, #72318733, #72318749
maximal normal subgroup of	#72318439, #72318733, #72318749

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389836);