#304516

order	512
nilpotency class	3
isoclinism	isoclinic class of #270875
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

814 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=eh,c^2=efh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi

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, 211712);