#6372173

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,22,17,21,23,30)(15,19,26,33,25,32,36,41)(16,27,29,40,28,20,34,35)(24,37,39,44,38,31,42,43);(1,2,3,4)(6,9)(10,11)(13,15,23,36,17,25,18,26)(14,19,30,41,21,32,22,33)(16,24,34,42,28,38,29,39)(20,31,40,44,27,37,35,43);(5,7,8,12)(6,10,9,11)(13,16,17,28)(14,20,21,27)(15,24,25,38)(18,29,23,34)(19,31,32,37)(22,35,30,40)(26,39,36,42)(33,43,41,44);(16,28)(20,27)(24,38)(29,34)(31,37)(35,40)(39,42)(43,44);(14,21)(19,32)(20,27)(22,30)(31,37)(33,41)(35,40)(43,44)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279369);