#7719047

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	#35236	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7626243);