#475902

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1136 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383098);