#7673537

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	#35088	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7854 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)(5,6)(7,11)(8,9)(10,12)(17,30)(21,34)(25,39)(28,41)(32,36)(33,40)(37,44)(42,43);(5,7)(6,10)(8,11)(9,12)(13,14,18,22)(15,23,26,19)(16,20,29,27)(17,21,30,34)(24,35,38,31)(25,36,39,32)(28,33,41,40)(37,43,44,42);(2,4)(5,8,9,6)(7,11,12,10)(13,14)(15,19)(16,27)(17,21)(18,22)(20,29)(23,26)(24,35)(25,32)(28,40)(30,34)(31,38)(33,41)(36,39)(37,43)(42,44);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,15,18,26)(14,19,22,23)(16,24,29,38)(17,25,30,39)(20,31,27,35)(21,32,34,36)(28,37,41,44)(33,42,40,43);(13,16)(14,20)(15,24)(17,28)(18,29)(19,31)(21,33)(22,27)(23,35)(25,37)(26,38)(30,41)(32,42)(34,40)(36,43)(39,44);(13,17)(14,21)(15,25)(16,28)(18,30)(19,32)(20,33)(22,34)(23,36)(24,37)(26,39)(27,40)(29,41)(31,42)(35,43)(38,44)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^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, 7580733);