#6370004

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

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

2520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=ci,e^d=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277200);