#363337

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	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1014 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=g,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, 270533);