#356947

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1638 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)(13,14,16,19,17,20,25,28)(15,21,22,26,23,27,24,18);(1,2)(3,4)(9,10,11,12);(5,6,8,7)(13,15,17,23)(14,18,20,26)(16,24,25,22)(19,27,28,21);(2,4)(5,7,8,6)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=fgi,f^2=i,g^2=i,b^a=be,c^a=cf,d^b=de,f^c=fi,g^c=gi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264143);