#393265

order	512
nilpotency class	3
isoclinism	isoclinic class of #32878
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

985 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ef,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,g^d=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300461);