#7145866

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
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×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2444 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:

(1,2,4,3)(7,11)(10,12)(13,14,18,22)(15,19,23,27)(16,21,17,20)(24,26,25,28);(2,3)(6,9)(7,11)(13,15,18,23)(14,19,22,27)(16,24,17,25)(20,28,21,26);(5,6)(7,10)(8,9)(11,12)(14,20)(15,23)(16,17)(19,26)(21,22)(27,28);(1,3,4,2)(6,9)(10,12)(13,16,18,17)(14,21,22,20)(15,24,23,25)(19,26,27,28);(5,7,8,11)(6,10,9,12)(15,23)(19,27)(24,25)(26,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7053062);