#357156

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×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

1330 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:

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

Polycyclic presentation:

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

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, 264352);