#7564153

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1940 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)(13,14,17,21)(15,23,26,19)(16,18,22,20)(24,25,28,27);(2,3)(6,10)(11,12)(13,15)(14,19)(16,28)(17,26)(18,25)(20,27)(21,23)(22,24);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,17,22)(14,20,21,18)(15,24,26,28)(19,25,23,27);(1,3,4,2)(14,21)(15,25)(18,20)(19,28)(23,24)(26,27);(5,8,9,7)(6,11,10,12)(14,21)(15,26)(16,22)(25,27)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=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, 7471349);