#362593

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,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, 269789);