#398345

order	512
nilpotency class	3
isoclinism	isoclinic class of #394378
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1583 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=fi,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,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, 305541);