#362592

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

869 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)(7,16,19,12)(8,20,22,34)(9,14)(10,15)(11,18,25,33)(13,21,28,35)(17,30)(23,27)(24,32)(26,31)(29,36);(1,2,3,4)(5,7)(6,11,15,25)(8,17)(9,18)(10,19)(12,14,16,27)(13,24,28,30)(20,29,34,26)(21,31)(22,32)(23,33)(35,36);(2,4)(5,7)(6,12)(8,17)(9,18)(10,19)(11,14)(13,26)(15,16)(20,24)(21,31)(22,32)(23,33)(25,27)(28,29)(30,34)(35,36);(5,8,9,21,10,22,23,35)(6,13,27,34,15,28,14,20)(7,17,18,31,19,32,33,36)(11,24,12,26,25,30,16,29)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,d^2=ef,e^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, 269788);