#7006331

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2126 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)(7,9)(8,10)(11,12)(13,14,20,26,18,23,19,24)(15,25,17,16,27,21,22,28);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,21)(16,24)(17,20)(18,27)(19,22)(23,25)(26,28);(5,7)(6,9)(8,12)(10,11)(13,16,18,28)(14,22,23,17)(15,26,27,24)(19,21,20,25);(1,3,4,2)(7,12)(9,11)(13,17,18,22)(14,16,23,28)(15,20,27,19)(21,24,25,26);(6,10)(9,11)(14,23)(15,27)(17,22)(24,26)

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6913527);