#7561372

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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

2330 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7468568);