#7630934

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×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7002 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)(6,7)(10,11)(13,14,19,23)(15,20,24,33)(16,17,21,30)(18,31,32,22)(25,26,34,37)(27,41,38,35)(28,42,36,29)(39,44,43,40);(5,6,9,7)(8,10,12,11)(15,24)(20,33)(25,34)(26,37)(27,38)(35,41)(39,43)(40,44);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,20)(16,25)(17,26)(18,27)(19,24)(21,34)(22,35)(23,33)(28,39)(29,40)(30,37)(31,41)(32,38)(36,43)(42,44);(5,8,9,12)(6,10,7,11)(13,16)(14,17)(15,25)(18,28)(19,21)(20,26)(22,29)(23,30)(24,34)(27,39)(31,42)(32,36)(33,37)(35,40)(38,43)(41,44);(13,17)(14,21)(15,26)(16,23)(18,29)(19,30)(20,34)(22,36)(24,37)(25,33)(27,40)(28,31)(32,42)(35,43)(38,44)(39,41);(13,18,19,32)(14,22,23,31)(15,27,24,38)(16,28,21,36)(17,29,30,42)(20,35,33,41)(25,39,34,43)(26,40,37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538130);