#423651

order	512
nilpotency class	3
isoclinism	isoclinic class of #417859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 330847);