#6384643

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	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1885 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)(10,11)(13,14,18,22)(15,19,17,20)(16,21,25,28)(23,26,24,27);(5,6,8,9)(7,11,12,10)(13,15,18,17)(14,19,22,20)(16,23,25,24)(21,26,28,27);(14,20)(15,17)(19,22)(21,27)(23,24)(26,28);(5,7,8,12)(6,10,9,11)(13,16,18,25)(14,21,22,28)(15,23,17,24)(19,26,20,27);(1,3,4,2)(13,15,18,17)(14,19,22,20)(16,24,25,23)(21,27,28,26)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ci,d^b=df,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, 6291839);