#7143804

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

2453 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)(14,18)(16,24)(19,25)(20,26)(21,28)(23,27);(6,9)(7,11)(13,14,16,18,17,20,24,26)(15,21,23,28,22,19,27,25);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,22)(14,19,20,21)(16,23,24,27)(18,25,26,28);(6,9)(10,12)(14,18)(15,22)(16,24)(19,28)(20,26)(21,25);(5,7,8,11)(6,10,9,12)(14,20)(15,22)(18,26)(23,27)

Polycyclic presentation:

2^9|b^2=f,c^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=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	#72307351, #72307390, #72307409
maximal normal subgroup of	#72307351, #72307390, #72307409

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7051000);