#385301

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

Aliases

No aliases for this group.

Subgroups

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

958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,d^2=f,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, 292497);