#7143545

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994171
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

2569 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)(16,19)(20,24)(21,25)(22,26)(27,28);(6,9)(7,11)(13,14,19,25,17,21,16,18)(15,20,26,28,23,27,22,24);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,23)(14,20,21,27)(16,22,19,26)(18,24,25,28);(6,9)(10,12)(14,18)(16,19)(20,24)(21,25)(22,26)(27,28);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,e^2=h,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	#72307365, #72307404, #72307429
maximal normal subgroup of	#72307365, #72307404, #72307429

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050741);