#7149213

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2005 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

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, 7056409);