#438213

order	512
nilpotency class	3
isoclinism	isoclinic class of #422233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

970 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)(7,18)(8,13)(9,26)(10,16)(11,17)(12,21)(14,28)(15,29)(19,34)(20,33)(22,31)(23,35)(24,36)(25,30)(27,32);(5,7,10,22,11,23,29,21)(6,12,16,18,17,31,15,35)(8,9,25,27,20,28,24,19)(13,32,30,14,33,34,36,26);(1,2,3,4)(5,8,11,20)(6,13,17,33)(7,19,23,27)(9,22,28,21)(10,24,29,25)(12,26,31,14)(15,30,16,36)(18,34,35,32);(5,9,10,27,11,28,29,19)(6,14,15,32,17,26,16,34)(7,20,21,25,23,8,22,24)(12,30,18,33,31,36,35,13)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345409);