#7715475

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×#3449	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7844 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)(5,6,9,7)(8,11,12,10)(14,19)(16,27)(17,30)(20,24)(25,35)(26,36)(34,41)(42,44);(1,2)(3,4)(14,19)(15,23)(21,28)(22,31)(25,35)(26,36)(34,41)(39,43);(6,7)(10,11)(13,14)(15,24)(16,28)(17,22)(18,19)(20,23)(21,27)(25,32)(26,38)(29,41)(30,31)(33,36)(34,40)(35,37)(39,42)(43,44);(5,7,9,6)(8,11,12,10)(13,15,18,23)(14,20,19,24)(16,25,27,35)(17,26,30,36)(21,32,28,37)(22,33,31,38)(29,39,40,43)(34,42,41,44);(5,8)(6,10)(7,11)(9,12)(13,16)(14,21)(15,25)(17,29)(18,27)(19,28)(20,32)(22,34)(23,35)(24,37)(26,39)(30,40)(31,41)(33,42)(36,43)(38,44);(13,17,18,30)(14,22,19,31)(15,26,23,36)(16,29,27,40)(20,33,24,38)(21,34,28,41)(25,39,35,43)(32,42,37,44)

Polycyclic presentation:

2^9|a^2=hi,d^2=h,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, 7622671);