#364558

order	512
nilpotency class	3
isoclinism	isoclinic class of #23872
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

2387 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)(5,6)(7,16)(8,13)(9,23)(10,14)(11,15)(12,20)(17,26)(18,33)(19,30)(21,28)(22,29)(24,27)(25,32)(31,34)(35,36);(1,3,4,2)(7,17)(9,10)(12,16)(14,23)(18,32)(19,20)(21,24)(25,31)(26,30)(27,28)(33,36)(34,35);(5,7)(6,12)(8,18)(9,17)(10,20)(11,19)(13,25)(14,30)(15,26)(16,23)(21,35)(22,34)(24,32)(27,31)(28,33)(29,36);(5,8,11,22)(6,13,15,29)(7,18,19,34)(9,24,10,21)(12,25,26,36)(14,28,23,27)(16,31,30,33)(17,32,20,35)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^c=fi,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, 271754);