#468888

order	512
nilpotency class	3
isoclinism	isoclinic class of #453960
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×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 376084);