#7595045

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	#59677	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1657 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,25)(19,27,29,23)(20,24,30,32)(22,31,28,26);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,29)(18,23,25,27)(20,22,30,28)(24,26,32,31);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,19,28,30,21,29,22,20)(18,24,31,23,25,32,26,27);(9,11,12,16)(10,14,13,15)(17,20,22,29,21,30,28,19)(18,23,26,24,25,27,31,32);(2,5)(6,8)(18,25)(23,27)(24,32)(26,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fh,d^2=g,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, 7502241);