#6385184

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, prime radical	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1767 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 6292380);