#7595043

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518936
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	#59650	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1721 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=fhi,d^2=gi,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=cgi,d^b=dh,g^b=gi,h^b=hi

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, 7502239);