#421760

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1596 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)(5,6,8,10)(7,11,12,9)(13,14,19,21,18,16,17,22)(15,23,24,27,25,20,26,28);(5,7,8,12)(6,9,10,11)(13,15,19,24,18,25,17,26)(14,20,21,28,16,23,22,27);(13,16,18,14)(15,23,25,20)(17,22,19,21)(24,28,26,27);(2,4)(7,12)(9,11)(13,17,18,19)(14,21,16,22)(15,24,25,26)(20,27,23,28)

Polycyclic presentation:

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

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, 328956);