#467241

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

956 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)(5,6,11,16)(7,13,8,20)(9,23)(10,28,17,15)(12,22,30,19)(14,26)(18,31)(21,35)(24,29)(25,32)(27,33)(34,36);(5,7,17,22,11,8,10,19)(6,12,28,20,16,30,15,13)(9,18,25,34,26,21,27,24)(14,29,32,35,23,36,33,31);(1,2,3,4)(5,8,11,7)(6,13,16,20)(9,24,26,34)(10,19,17,22)(12,28,30,15)(14,29,23,36)(18,25,21,27)(31,33,35,32);(5,9)(6,14)(7,18)(8,21)(10,27)(11,26)(12,29)(13,31)(15,33)(16,23)(17,25)(19,24)(20,35)(22,34)(28,32)(30,36)

Polycyclic presentation:

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

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, 374437);