#7062707

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
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×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2170 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)(16,25)(17,19)(18,24)(20,26)(22,28)(23,27);(5,6)(7,11)(8,10)(9,12)(13,15)(14,18)(16,23)(17,22)(19,27)(20,28)(21,25)(24,26);(2,4)(5,7,8,12)(6,9,10,11)(15,22)(18,27)(21,28)(23,24);(7,12)(9,11)(15,23)(16,26)(18,28)(20,25)(21,27)(22,24);(6,10)(9,11)(14,19)(15,22)(20,25)(23,24)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,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=dhi,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314102, #72314130, #72314210
maximal normal subgroup of	#72314102, #72314130, #72314210

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6969903);