#7630605

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	#35634	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6964 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)(5,6,8,10)(7,9,11,12)(17,29)(22,31)(25,39)(28,40)(33,36)(34,41)(38,44)(42,43);(1,2,3,4)(14,19)(16,27)(20,23)(22,31)(24,37)(28,40)(33,36)(38,44);(5,6)(7,9)(8,10)(11,12)(13,14,18,19)(15,23,26,20)(16,21,27,30)(17,22,29,31)(24,35,37,32)(25,36,39,33)(28,34,40,41)(38,43,44,42);(13,15,18,26)(14,20,19,23)(16,24,27,37)(17,25,29,39)(21,32,30,35)(22,33,31,36)(28,38,40,44)(34,42,41,43);(13,16)(14,21)(15,24)(17,28)(18,27)(19,30)(20,32)(22,34)(23,35)(25,38)(26,37)(29,40)(31,41)(33,42)(36,43)(39,44);(5,7,8,11)(6,9,10,12)(13,17)(14,22)(15,25)(16,28)(18,29)(19,31)(20,33)(21,34)(23,36)(24,38)(26,39)(27,40)(30,41)(32,42)(35,43)(37,44)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=i,f^2=h,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, 7537801);