#396102

order	512
nilpotency class	3
isoclinism	isoclinic class of #396101
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1424 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)(7,16,18,11)(8,12)(9,15)(10,14)(13,19,29,34)(17,24,32,25)(20,28)(21,27)(22,31)(23,30)(26,33,36,35);(1,2)(3,4)(5,7)(6,11)(8,17)(9,19)(10,18)(12,25)(13,15)(14,16)(20,33)(21,32)(22,35)(23,34)(24,27)(26,28)(29,30)(31,36);(2,4)(5,8,10,21)(6,12,14,27)(7,17,18,32)(9,22,23,20)(11,24,16,25)(13,26,29,36)(15,31,30,28)(19,35,34,33);(5,7)(6,13,14,29)(8,17)(9,19)(10,18)(11,15,16,30)(12,26,27,36)(20,33)(21,32)(22,35)(23,34)(24,31,25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303298);