#368234

order	512
nilpotency class	3
isoclinism	isoclinic class of #23998
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1421 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=eg,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,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, 275430);