#391206

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,12,18,11,17,10,16)(7,19,22,13,8,14,21,29)(9,15,25,32,26,33,27,34)(20,35,36,28,23,30,24,31);(1,2)(3,4)(5,7,11,8)(6,13,17,29)(9,24,26,36)(10,22,12,21)(14,18,19,16)(15,30,33,35)(20,25,23,27)(28,34,31,32);(2,4)(5,8,11,7)(6,14,17,19)(9,24,26,36)(10,21,12,22)(13,16,29,18)(15,31,33,28)(20,25,23,27)(30,34,35,32);(5,9)(6,15)(7,20)(8,23)(10,27)(11,26)(12,25)(13,28)(14,30)(16,34)(17,33)(18,32)(19,35)(21,24)(22,36)(29,31)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei

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, 298402);