#368087

order	512
nilpotency class	3
isoclinism	isoclinic class of #23897
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1639 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)(7,15)(8,12)(9,21)(10,14)(11,18)(13,23)(16,31)(17,29)(19,28)(20,27)(22,26)(24,34)(25,33)(30,32)(35,36);(1,3,4,2)(11,24)(15,29)(25,36)(30,31);(5,7)(6,11,14,24)(8,16)(9,18)(10,17)(12,25,27,36)(13,15,21,29)(19,33)(20,32)(22,35)(23,34)(26,31,28,30);(5,8,10,20)(6,12,14,27)(7,16,17,32)(9,22,23,19)(11,25,24,36)(13,28,21,26)(15,30,29,31)(18,35,34,33)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,e^c=ei,f^d=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, 275283);