#7054568

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2306 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)(5,6)(7,9)(8,10)(11,12)(14,19)(16,17)(20,21)(22,23)(24,28)(25,27);(5,6)(7,11)(8,10)(9,12)(13,14,18,23)(15,24,26,20)(16,19,17,22)(21,27,28,25);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,27)(17,25)(18,26)(19,21)(22,28)(23,24);(7,12)(9,11)(13,15)(14,21)(16,25)(17,27)(18,26)(19,20)(22,24)(23,28);(6,10)(9,11)(13,16,18,17)(14,22,23,19)(15,25,26,27)(20,21,24,28)

Polycyclic presentation:

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

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, 6961764);