#7596608

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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	#59824	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1574 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:

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

Polycyclic presentation:

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

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, 7503804);