#6384478

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
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×#546	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2163 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

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,c^b=ci,d^b=df,e^b=ei,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291674);