#366451

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

958 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273647);