#494955

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

561 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)(5,6,11,16,10,17,30,35,12,18,32,14,31,15,9,28)(7,19,22,25,23,27,33,34,24,36,20,29,21,8,13,26);(5,7,11,22,10,23,30,33,12,24,32,20,31,21,9,13)(6,8,16,26,17,19,35,25,18,27,14,34,15,36,28,29);(2,4)(5,8)(6,13)(7,16)(9,29)(10,19)(11,26)(12,27)(14,24)(15,20)(17,22)(18,33)(21,28)(23,35)(25,30)(31,36)(32,34);(5,9,10,11,12,30,31,32)(6,14,15,35,18,16,17,28)(7,20,21,33,24,22,23,13)(8,25,19,34,27,29,36,26)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,d^b=dh,e^b=ei,d^c=di,g^d=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402151);