#7195647

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2498 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=f,e^2=hi,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^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7102843);