#396357

order	512
nilpotency class	3
isoclinism	isoclinic class of #395181
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23445	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1505 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)(8,20,22,27)(9,14)(10,15)(11,18)(12,21,28,34)(13,23)(17,24,31,35)(19,25)(26,32)(29,30,33,36);(1,2)(3,4)(5,7,9,18,10,19,23,32)(6,11,13,16,15,26,14,25)(8,17,21,30,22,31,34,36)(12,24,27,29,28,35,20,33);(2,4)(5,6)(7,11)(8,12)(9,13)(10,15)(14,23)(16,18)(17,24)(19,26)(20,34)(21,27)(22,28)(25,32)(29,30)(31,35)(33,36);(5,8,9,21,10,22,23,34)(6,12,13,27,15,28,14,20)(7,17,18,30,19,31,32,36)(11,24,16,29,26,35,25,33)

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,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, 303553);