#6384202

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×#553	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2144 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:

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

Polycyclic presentation:

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

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, 6291398);