#499513

order	512
nilpotency class	4
isoclinism	isoclinic class of #499266
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

565 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320223
maximal normal subgroup of	#72320223

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406709);