#356953

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
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

1614 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)(13,14,17,20,16,18,24,27)(15,21,23,26,22,28,25,19);(1,2)(3,4)(9,10,11,12)(14,18)(19,26)(20,27)(21,28);(5,6,8,7)(13,15,16,22)(14,19,18,26)(17,25,24,23)(20,28,27,21);(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=dei,f^c=fi,g^c=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, 264149);