#7565307

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	C₂×D₄²	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1985 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)(8,12)(9,10)(13,14)(15,22)(16,18)(17,21)(19,25)(20,28)(23,24)(26,27);(2,3)(6,10)(11,12)(13,15,17,25)(14,19,21,22)(16,27,28,23)(18,24,20,26);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,23)(17,28)(18,21)(19,24)(22,26)(25,27);(1,3,4,2)(5,7,9,8)(6,11,10,12)(14,21)(15,24)(18,20)(19,23)(22,27)(25,26);(5,8,9,7)(6,11,10,12)(14,21)(16,28)(19,22)(23,27)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=fg,e^2=fi,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,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7472503);