#7596941

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2189 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,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, 7504137);