#7143592

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
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

2401 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)(13,14,19,16)(15,22,23,24)(17,21,18,28)(20,25,27,26);(6,9)(7,11)(13,15,18,26,19,23,17,25)(14,20,21,22,16,27,28,24);(2,4)(5,6)(7,10)(8,9)(11,12)(14,16)(15,23)(21,28)(25,26);(6,9)(10,12)(13,16,19,14)(15,24,23,22)(17,28,18,21)(20,26,27,25);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,16,28)(15,25,23,26)(20,22,27,24)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283141, #72283296, #72284087
maximal normal subgroup of	#72283141, #72283296, #72284087

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050788);