#232155

order	512
nilpotency class	3
isoclinism	isoclinic class of #232153
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#560
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

397 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,11,14,7)(4,15,19,30)(5,9)(6,10)(8,18,24,28)(12,23)(13,25,20,22)(16,21)(17,26)(27,31)(29,32)(33,34)(35,45)(36,42,52,41)(37,53,54,61)(38,55)(39,44)(40,48)(43,51)(46,59,63,60)(47,57,50,64)(49,62)(56,58);(1,3,6,14)(2,7,10,11)(4,16,19,31)(5,20,17,13)(8,23,24,32)(9,25,26,22)(12,28,29,18)(15,27,30,21)(33,35,39,49)(34,40,44,58)(36,50,52,47)(37,46,54,63)(38,56,51,48)(41,59,42,60)(43,62,55,45)(53,64,61,57);(1,4,17,28,6,19,5,18)(2,8,9,15,10,24,26,30)(3,12,20,31,14,29,13,16)(7,21,22,32,11,27,25,23)(33,36,51,54,39,52,38,37)(34,41,43,53,44,42,55,61)(35,46,56,47,49,63,48,50)(40,57,45,60,58,64,62,59);(2,9,10,26)(7,22,11,25)(8,15,24,30)(21,32,27,23)(33,37,38,52,39,54,51,36)(34,42,43,61,44,41,55,53)(35,47,48,46,49,50,56,63)(40,57,45,60,58,64,62,59)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=efh,d^2=e,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cef,d^b=de,e^b=ei,f^b=fhi,g^b=ghi,g^c=gh,g^d=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139351);