#7618330

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 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)(3,7)(4,8)(5,6)(10,14)(11,12)(17,18,23,26)(19,28,29,24)(20,22,27,31)(21,25,30,32);(2,6)(7,8)(9,10)(11,16)(12,15)(13,14)(17,19,23,29)(18,24,26,28)(20,25,27,32)(21,31,30,22);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20)(18,22)(19,25)(21,24)(23,27)(26,31)(28,30)(29,32);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,16,14,15)(17,21,22,19,23,30,31,29)(18,25,27,28,26,32,20,24);(2,6)(7,8)(9,12,13,11)(10,15,14,16)(17,22,23,31)(18,20,26,27)(19,21,29,30)(24,25,28,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=fhi,e^2=gi,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7525526);