#7143556

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

2454 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)(13,14,19,22)(15,23,26,27)(16,17,21,18)(20,24,28,25);(6,9)(7,11)(13,15,18,25,19,26,17,24)(14,20,21,23,22,28,16,27);(2,4)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12)(13,16)(14,17)(15,20)(18,22)(19,21)(23,24)(25,27)(26,28);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,22,16)(15,24,26,25)(20,23,28,27)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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,f^d=fi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282583, #72283296, #72283984
maximal normal subgroup of	#72282583, #72283296, #72283984

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050752);