#361536

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

719 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:

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

Polycyclic presentation:

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

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, 268732);