#499511

order	512
nilpotency class	4
isoclinism	isoclinic class of #499263
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×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

575 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:

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

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,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, 406707);