#364520

order	512
nilpotency class	3
isoclinism	isoclinic class of #23882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2401 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)(5,6,11,16)(7,12,9,14)(8,13,21,30)(10,15,25,34)(17,29,20,26)(18,27,23,32)(19,36,35,28)(22,31,24,33);(1,3,4,2)(5,7,11,9)(6,12,16,14)(8,19,21,35)(10,23,25,18)(13,28,30,36)(15,32,34,27)(17,24,20,22)(26,33,29,31);(5,8)(6,13)(7,17)(9,20)(10,24)(11,21)(12,26)(14,29)(15,33)(16,30)(18,35)(19,23)(22,25)(27,36)(28,32)(31,34);(5,9,11,7)(6,14,16,12)(8,20,21,17)(10,23,25,18)(13,29,30,26)(15,32,34,27)(19,22,35,24)(28,31,36,33)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,g^b=gi,e^c=ei,g^c=gi,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, 271716);