#476995

order	512
nilpotency class	3
isoclinism	isoclinic class of #476112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1070 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)(7,13)(8,14)(9,12)(10,15)(11,16)(17,25)(18,33)(19,28)(20,29)(21,32)(22,31)(23,30)(24,26)(27,34)(35,36);(1,3,4,2)(6,12)(7,17)(8,21)(9,10)(14,30)(15,16)(18,34)(19,20)(23,24)(25,28)(27,31)(33,36);(2,3)(5,7)(6,13)(8,22)(9,17)(10,19)(11,20)(12,25)(14,31)(15,28)(16,29)(18,21)(23,35)(24,34)(26,27)(30,36)(32,33);(2,3)(5,8)(6,14)(7,18)(9,24)(10,21)(11,23)(12,26)(13,27)(15,32)(16,30)(17,22)(19,35)(20,34)(25,36)(28,31)(29,33)

Polycyclic presentation:

2^9|a^2=f,b^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384191);