#6389669

order	512
nilpotency class	3
isoclinism	isoclinic class of #62295
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×#3171	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2054 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)(5,6,10,12)(7,8,11,9)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,28)(25,27);(1,3,4,2)(13,15)(14,19)(16,23)(17,25)(18,24)(20,27)(21,22)(26,28);(5,7,9,6,10,11,8,12)(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,27,22,28);(5,8,10,9)(6,11,12,7)(15,24)(19,22)(23,25)(27,28);(5,8,10,9)(6,11,12,7)(13,17,18,16)(14,20,21,26)(15,25,24,23)(19,27,22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296865);