#6370229

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

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	C₂²×#41	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2328 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277425);