#6380245

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2420 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)(6,9)(10,12)(13,14)(15,22)(16,20)(17,21)(18,26)(19,25)(23,27)(24,28);(5,6)(7,10)(8,9)(11,12)(13,14,17,21)(15,19,24,27)(16,26,18,20)(22,23,28,25);(5,7,8,11)(6,10,9,12)(13,15,16,23,17,24,18,25)(14,19,26,28,21,27,20,22);(6,9)(10,12)(13,14)(15,19)(16,26)(17,21)(18,20)(22,25)(23,28)(24,27);(13,16,17,18)(14,20,21,26)(15,23,24,25)(19,22,27,28)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fg,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287441);