#464057

order	512
nilpotency class	3
isoclinism	isoclinic class of #456505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1471 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,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, 371253);