#7062940

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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
prime radical	#24441	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1893 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,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, 6970136);