#7715498

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(2,4)(6,7)(10,13)(14,16)(17,18,22,26)(19,27,28,23)(20,24,31,38)(21,34,35,25)(29,39,32,36)(30,37,42,41)(33,46,44,40)(43,47,45,48);(1,2,5,4)(3,6,8,7)(9,10,12,13)(11,14,15,16)(19,28)(20,31)(23,27)(24,38)(30,42)(33,44)(37,41)(40,46);(1,2,5,4)(3,7,8,6)(17,19,22,28)(18,23,26,27)(20,32,31,29)(21,30,35,42)(24,39,38,36)(25,37,34,41)(33,45,44,43)(40,48,46,47);(1,3,5,8)(2,6,4,7)(17,20,22,31)(18,24,26,38)(19,29,28,32)(21,33,35,44)(23,36,27,39)(25,40,34,46)(30,43,42,45)(37,47,41,48);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,39)(30,41)(31,38)(32,36)(33,40)(34,35)(37,42)(43,48)(44,46)(45,47);(17,21,22,35)(18,25,26,34)(19,30,28,42)(20,33,31,44)(23,37,27,41)(24,40,38,46)(29,43,32,45)(36,47,39,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622694);