#232725

order	512
nilpotency class	3
isoclinism	isoclinic class of #232724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60958	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,g^b=gh,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139921);