#7630859

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 44 points:

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

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538055);