#356340

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, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1254 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)(5,6,7,8)(9,10,12,11)(13,14,19,15,18,21,17,23)(16,20,25,22,26,28,27,24);(10,11)(13,15,19,21,18,23,17,14)(16,24,25,20,26,22,27,28);(1,2,3,4)(13,16)(14,20)(15,22)(17,27)(18,26)(19,25)(21,28)(23,24);(9,11,12,10)(13,17,18,19)(14,15,21,23)(16,25,26,27)(20,24,28,22)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^b=de,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, 263536);