#7674631

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

7014 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,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, 7581827);