#406369

order	512
nilpotency class	3
isoclinism	isoclinic class of #394183
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=f,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313565);