#7146093

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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

2348 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)(13,14,19,22,18,17,16,21)(15,23,26,20,25,28,24,27);(6,9)(7,11)(13,15,18,25)(14,20,17,27)(16,26,19,24)(21,28,22,23);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,19)(14,21,17,22)(15,24,25,26)(20,23,27,28);(6,9)(10,12)(13,17,19,21,18,14,16,22)(15,23,26,20,25,28,24,27);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,22,17,21)(15,24,25,26)(20,28,27,23)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282638, #72283321, #72283956
maximal normal subgroup of	#72282638, #72283321, #72283956

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7053289);