#460332

order	512
nilpotency class	3
isoclinism	isoclinic class of #460330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
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

1164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,10,17,12,18,11,16)(7,19,22,13,24,38,23,39)(8,25,28,43,30,42,29,14)(9,31,33,46,35,45,34,15)(20,36,26,40,50,55,49,63)(21,37,52,60,54,65,53,47)(27,57,32,44,59,41,58,56)(48,66,51,68,61,64,67,62);(5,7,10,22,12,24,11,23)(6,13,16,19,18,39,17,38)(8,26,28,50,30,49,29,20)(9,21,33,52,35,54,34,53)(14,40,42,36,43,63,25,55)(15,37,45,47,46,65,31,60)(27,51,32,61,59,67,58,48)(41,66,44,62,57,64,56,68);(5,8,12,30)(6,14,18,43)(7,20,24,50)(9,32,35,58)(10,29,11,28)(13,36,39,55)(15,44,46,56)(16,25,17,42)(19,40,38,63)(21,51,54,67)(22,49,23,26)(27,34,59,33)(31,57,45,41)(37,64,65,66)(47,62,60,68)(48,53,61,52);(2,4)(5,9,11,34,12,35,10,33)(6,15,17,31,18,46,16,45)(7,21,23,53,24,54,22,52)(8,27,29,58,30,59,28,32)(13,37,38,60,39,65,19,47)(14,41,25,56,43,57,42,44)(20,48,49,67,50,61,26,51)(36,62,40,66,55,68,63,64)

Polycyclic presentation:

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

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, 367528);