#7003376

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994171
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

2120 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)(13,14)(15,21)(16,23)(17,22)(18,20)(19,25)(24,28)(26,27);(5,6)(7,11)(8,10)(9,12)(13,15,16,26,18,24,17,25)(14,19,22,28,20,27,23,21);(2,4)(5,7)(6,9)(8,12)(10,11)(14,20)(15,24)(22,23)(25,26);(7,12)(9,11)(13,15)(14,21)(16,25)(17,26)(18,24)(19,23)(20,28)(22,27);(6,10)(9,11)(13,16,18,17)(14,22,20,23)(15,25,24,26)(19,21,27,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,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=eh,d^c=dh,f^d=fi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280939, #72283980
maximal normal subgroup of	#72280939, #72283980

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6910572);