#391866

order	512
nilpotency class	3
isoclinism	isoclinic class of #232147
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3153	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1376 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299062);