#304382

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ei,c^2=ef,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211578);