#7143763

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2335 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)(7,11)(10,12)(13,14,19,25)(15,22,21,16)(17,24,18,23)(20,27,28,26);(6,9)(7,11)(13,15,17,27,19,21,18,26)(14,20,23,16,25,28,24,22);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,22)(14,21,25,15)(17,28,18,20)(23,26,24,27);(6,9)(10,12)(13,15)(14,22)(16,25)(17,26)(18,27)(19,21)(20,24)(23,28);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,25,24)(15,26,21,27)(16,20,22,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,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, 7050959);