#367096

order	512
nilpotency class	3
isoclinism	isoclinic class of #24808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1446 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274292);