#7149181

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

2012 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)(13,14,20,28,19,22,18,21)(15,23,24,27,17,16,25,26);(1,2,3,4)(6,9)(7,11)(13,15)(14,16)(17,19)(18,25)(20,24)(21,27)(22,23)(26,28);(5,6)(7,10)(8,9)(11,12)(13,16,19,23)(14,15,22,17)(18,26,20,27)(21,24,28,25);(2,4)(6,9)(10,12)(13,17,19,15)(14,16,22,23)(18,24,20,25)(21,27,28,26);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,21,22,28)(15,24,17,25)(16,26,23,27)

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=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056377);