#455257

order	512
nilpotency class	3
isoclinism	isoclinic class of #454372
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1280 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)(5,6,11,16)(7,17,21,34)(8,22,24,31)(9,14,23,13)(10,26,25,15)(12,27,29,20)(18,28,36,30)(19,33,35,32);(5,7,10,20,11,21,25,27)(6,12,15,17,16,29,26,34)(8,18,23,35,24,36,9,19)(13,30,22,32,14,28,31,33);(2,4)(5,8)(6,13)(7,18)(9,10)(11,24)(12,28)(14,16)(15,31)(17,32)(19,20)(21,36)(22,26)(23,25)(27,35)(29,30)(33,34);(5,9,25,24,11,23,10,8)(6,14,26,22,16,13,15,31)(7,19,27,36,21,35,20,18)(12,28,34,32,29,30,17,33)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362453);