#357119

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264315);