#385484

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1410 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ef,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=dei,d^c=di

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, 292680);