#212806

order	512
nilpotency class	3
isoclinism	isoclinic class of #32929
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

774 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=f,d^2=gh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120002);