#6380403

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
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×#121	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1926 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)(13,14,18,22)(15,23,26,20)(16,17,21,19)(24,25,28,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,24)(17,27)(18,26)(19,25)(21,28)(22,23);(13,16,19,14,18,21,17,22)(15,24,25,20,26,28,27,23);(6,9)(10,12)(13,17,18,19)(14,21,22,16)(15,25,26,27)(20,24,23,28);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,21,22,16)(15,25,26,27)(20,24,23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287599);