#457105

order	512
nilpotency class	3
isoclinism	isoclinic class of #457102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

1092 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)(5,6,12,16,11,18,10,17)(7,19,34,37,23,46,22,13)(8,14,28,40,27,42,26,41)(9,29,33,45,32,44,31,15)(20,47,55,35,50,38,24,56)(21,51,60,66,54,36,53,48)(25,43,58,64,57,63,30,39)(49,61,68,62,59,65,52,67);(5,7,11,23)(6,13)(8,24,27,55)(9,21,32,54)(10,22,12,34)(14,38)(15,36)(16,19)(17,46)(18,37)(20,26,50,28)(25,52,57,68)(29,48)(30,59,58,49)(31,53,33,60)(35,41)(39,62)(40,56)(42,47)(43,65)(44,66)(45,51)(61,63)(64,67);(5,8,11,27)(6,14,18,42)(7,20,23,50)(9,30,32,58)(10,28,12,26)(13,35,37,56)(15,43,45,63)(16,41,17,40)(19,47,46,38)(21,52,54,68)(22,55,34,24)(25,33,57,31)(29,39,44,64)(36,62,51,67)(48,61,66,65)(49,60,59,53);(2,4)(5,9,12,33,11,32,10,31)(6,15,17,44,18,45,16,29)(7,21,34,60,23,54,22,53)(8,25,28,58,27,57,26,30)(13,36,46,66,37,51,19,48)(14,39,41,63,42,64,40,43)(20,49,55,68,50,59,24,52)(35,61,47,67,56,65,38,62)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=i,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,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, 364301);