#381888

order	512
nilpotency class	3
isoclinism	isoclinic class of #219508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1173 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289084);