#100507

order	512
nilpotency class	3
isoclinism	isoclinic class of #100251
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

461 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=gh,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7703);