#6993928

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993915
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 ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2323 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)(3,4)(5,6)(7,9)(8,10)(11,12)(14,19)(15,17)(20,22)(21,27)(23,25)(24,28);(5,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,19,17,20)(16,24,26,27)(21,25,28,23);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,17)(14,20,22,19)(16,25,26,23)(21,27,28,24);(7,12)(9,11)(13,16)(14,21)(15,23)(17,25)(18,26)(19,27)(20,24)(22,28);(6,10)(9,11)(13,15,18,17)(14,20,22,19)(16,23,26,25)(21,24,28,27)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6901124);