#7674636

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	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6646 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)(5,6,9,10)(7,11,12,8)(14,19)(17,30)(20,24)(21,27)(26,36)(28,41)(32,37)(39,43);(1,2,3,4)(14,19)(15,23)(21,27)(22,31)(25,35)(26,36)(34,40)(39,43);(5,7,9,12)(6,8,10,11)(13,14,18,19)(15,24,23,20)(16,27,29,21)(17,22,30,31)(25,32,35,37)(26,38,36,33)(28,40,41,34)(39,42,43,44);(5,8)(6,7)(9,11)(10,12)(13,15)(14,20)(16,25)(17,26)(18,23)(19,24)(21,32)(22,33)(27,37)(28,39)(29,35)(30,36)(31,38)(34,42)(40,44)(41,43);(13,16,18,29)(14,21,19,27)(15,25,23,35)(17,28,30,41)(20,32,24,37)(22,34,31,40)(26,39,36,43)(33,42,38,44);(13,17,18,30)(14,22,19,31)(15,26,23,36)(16,28,29,41)(20,33,24,38)(21,34,27,40)(25,39,35,43)(32,42,37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581832);