#7143658

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993979
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 ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2403 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)(13,14,18,16)(15,21,22,28)(17,25,26,20)(19,27,24,23);(6,9)(7,11)(13,15,17,23,18,22,26,27)(14,19,20,21,16,24,25,28);(2,4)(5,6)(7,10)(8,9)(11,12)(14,16)(15,22)(20,25)(23,27);(6,9)(10,12)(13,16,18,14)(15,21,22,28)(17,20,26,25)(19,27,24,23);(5,7,8,11)(6,10,9,12)(15,22)(19,24)(21,28)(23,27)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283039, #72283296, #72284204
maximal normal subgroup of	#72283039, #72283296, #72284204

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050854);