#7143754

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

2419 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)(13,14,19,23)(15,16,21,27)(17,22,18,28)(20,24,26,25);(6,9)(7,11)(13,15,18,25,19,21,17,24)(14,20,22,16,23,26,28,27);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,23)(17,20)(18,26)(19,27)(22,24)(25,28);(6,9)(10,12)(13,14,19,23)(15,16,21,27)(17,22,18,28)(20,24,26,25);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,23,28)(15,24,21,25)(16,26,27,20)

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=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.

Relationships

maximal subgroup of	#72284010, #72284285, #72284702
maximal normal subgroup of	#72284010, #72284285, #72284702

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050950);