#7608213

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=gh,d^2=g,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7515409);