#245459

order	512
nilpotency class	3
isoclinism	isoclinic class of #245447
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60832	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1092 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:

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

Polycyclic presentation:

2^9|a^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152655);