#7143028

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993979
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2272 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)(16,25)(19,26)(20,21)(22,27)(24,28);(6,9)(7,11)(13,14,17,21)(15,22,23,19)(16,20,25,18)(24,26,28,27);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,24)(17,23)(18,26)(20,27)(21,22)(25,28);(6,9)(10,12)(14,20)(15,23)(16,25)(18,21)(19,26)(22,27);(5,7,8,11)(6,10,9,12)(14,21)(18,20)(19,22)(26,27)

Polycyclic presentation:

2^9|b^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=df,e^b=eh,f^b=fi,d^c=dh,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307419, #72307449, #72308077
maximal normal subgroup of	#72307419, #72307449, #72308077

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050224);