#7630813

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	#34633	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7706 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)(6,11)(7,12)(13,14)(15,19)(16,20)(17,21)(18,25)(22,32)(23,28)(24,31)(26,33)(27,37)(29,35)(30,38)(34,40)(36,42)(39,44)(41,43);(5,6,10,11)(7,8,12,9)(14,15)(20,23)(21,24)(22,25)(33,35)(34,37)(36,38)(43,44);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,15,19,14)(16,23,28,20)(17,24,31,21)(18,25,32,22)(26,35,29,33)(27,37,40,34)(30,38,42,36)(39,44,41,43);(5,8,10,9)(6,12,11,7)(13,16,19,28)(14,20,15,23)(17,29,31,26)(18,27,32,40)(21,35,24,33)(22,34,25,37)(30,41,42,39)(36,44,38,43);(5,9,10,8)(6,7,11,12)(13,17,19,31)(14,21,15,24)(16,26,28,29)(18,30,32,42)(20,33,23,35)(22,36,25,38)(27,39,40,41)(34,43,37,44);(13,18)(14,22)(15,25)(16,27)(17,30)(19,32)(20,34)(21,36)(23,37)(24,38)(26,39)(28,40)(29,41)(31,42)(33,43)(35,44)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=g,e^2=g,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, 7538009);