#397748

order	512
nilpotency class	3
isoclinism	isoclinic class of #397514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1252 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)(3,4)(5,6,10,20,11,14,23,8)(7,15,17,12,18,28,32,25)(9,19,21,13,22,33,34,26)(16,29,30,27,31,24,35,36);(5,7)(6,12,14,25)(8,15,20,28)(9,16)(10,17)(11,18)(13,24,26,29)(19,27,33,36)(21,30)(22,31)(23,32)(34,35);(5,8,10,14,11,20,23,6)(7,15,17,25,18,28,32,12)(9,19,21,26,22,33,34,13)(16,27,30,29,31,36,35,24);(2,4)(5,9,10,21,11,22,23,34)(6,13,8,19,14,26,20,33)(7,16,17,30,18,31,32,35)(12,24,15,27,25,29,28,36)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=e,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=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, 304944);