#7620604

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,6)(3,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,27)(19,21,26,24)(20,23,28,25)(29,32,31,30);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,26)(18,24,27,21)(20,31,28,29)(23,32,25,30);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,28)(18,25,27,23)(19,29,26,31)(21,30,24,32);(1,4,5,3)(2,8,6,7)(10,13)(14,16)(17,21,23,29,22,24,25,31)(18,26,28,32,27,19,20,30);(2,6)(7,8)(9,11,12,15)(10,14,13,16)(18,27)(20,28)(21,24)(29,31)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fgh,e^2=gi,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, 7527800);