#398453

order	512
nilpotency class	3
isoclinism	isoclinic class of #395252
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1624 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:

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

Polycyclic presentation:

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

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, 305649);