#7630657

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
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 7537853);