#366703

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#3018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2238 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)(7,10)(8,17,22,19)(9,18)(11,15)(12,16)(13,21,28,26)(14,23)(20,33,36,27)(24,29)(25,30)(31,35,32,34);(1,2,3,4)(5,7,10,16,12,15,11,6)(8,19,21,13,22,17,26,28)(9,18,25,30,24,29,23,14)(20,32,35,27,36,31,34,33);(2,4)(5,8)(6,13)(7,17)(9,20)(10,21)(11,26)(12,22)(14,27)(15,19)(16,28)(18,31)(23,34)(24,36)(25,35)(29,32)(30,33);(5,9)(6,14)(7,18)(8,20)(10,25)(11,23)(12,24)(13,27)(15,29)(16,30)(17,31)(19,32)(21,35)(22,36)(26,34)(28,33)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273899);