#7611541

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
prime radical	#45263	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2104 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)(7,9)(8,10)(11,12)(15,22)(19,27)(23,28)(24,26);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,14)(15,23)(16,20)(17,25)(18,21)(19,24)(22,28)(26,27);(6,10)(7,11)(13,14)(15,19)(16,25)(17,20)(18,21)(22,27)(23,24)(26,28);(6,10)(9,12)(13,15,17,24,18,22,16,26)(14,19,20,23,21,27,25,28);(1,3,4,2)(13,16,18,17)(14,20,21,25)(15,24,22,26)(19,28,27,23)

Polycyclic presentation:

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

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, 7518737);