#7715441

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

6740 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)(13,14)(15,23)(16,27)(17,31)(18,19)(20,24)(21,30)(22,32)(25,33)(26,34)(28,35)(29,36)(37,38)(39,44)(40,42)(41,43);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(14,19)(15,24)(21,27)(22,31)(25,28)(26,38)(36,40)(39,41);(6,10)(8,11)(13,15)(14,20)(16,28)(17,26)(18,24)(19,23)(21,35)(22,34)(25,30)(27,33)(29,41)(31,37)(32,38)(36,44)(39,42)(40,43);(5,8)(6,7)(9,11)(10,12)(13,14)(15,23)(16,21)(17,22)(18,19)(20,24)(25,35)(26,37)(27,30)(28,33)(29,36)(31,32)(34,38)(39,44)(40,42)(41,43);(5,7)(6,8)(9,12)(10,11)(13,16)(14,21)(15,25)(17,29)(18,30)(19,27)(20,33)(22,36)(23,35)(24,28)(26,39)(31,40)(32,42)(34,43)(37,44)(38,41);(13,17,18,32)(14,22,19,31)(15,26,24,38)(16,29,30,42)(20,34,23,37)(21,36,27,40)(25,39,28,41)(33,43,35,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,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, 7622637);