#6371006

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
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	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4166 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ci,e^c=ei,e^d=ei

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, 6278202);