#424388

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331584);