#6370226

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

2352 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,g^2=i,b^a=bf,d^a=df,c^b=cfi,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, 6277422);