#127079

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×Q₈	C₂³
socle	C₂²	C₂×#342
prime radical	C₂⁴	#95
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=h,f^2=hi,b^a=bd,d^a=dg,f^a=fi,g^a=gi,c^b=ci,e^b=egi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34275);