#7596903

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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

2164 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)(14,20)(15,23)(16,18)(17,21)(22,26)(24,28);(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,18,20,19,22,16,26)(15,24,25,17,23,21,27,28);(1,2,4,3)(6,9)(10,11)(13,15,19,23)(14,21,22,24)(16,27,18,25)(17,20,28,26);(1,3,4,2)(5,7,8,12)(6,10,9,11)(13,16,19,18)(14,20,22,26)(15,25,23,27)(17,24,28,21);(13,17,18,21,19,28,16,24)(14,15,20,25,22,23,26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7504099);