#455905

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1053 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ei,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^d=ei,g^d=gi

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, 363101);