#7939939

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

6545 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)(5,6,10,7)(8,13,16,11)(9,12,17,14)(15,19,20,18)(21,22,26,30)(23,31,34,27)(24,33,35,28)(25,29,36,32);(1,2)(3,4)(21,23,26,34)(22,27,30,31)(24,32,35,29)(25,33,36,28);(6,7)(11,13)(12,14)(18,19)(21,24,26,35)(22,28,30,33)(23,29,34,32)(25,27,36,31);(5,7,10,6)(8,13,16,11)(9,14,17,12)(15,19,20,18)(21,24)(22,28)(23,32)(25,27)(26,35)(29,34)(30,33)(31,36);(5,8)(6,11)(7,13)(9,15)(10,16)(12,18)(14,19)(17,20)(21,22)(23,27)(24,28)(25,32)(26,30)(29,36)(31,34)(33,35);(5,9,10,17)(6,12,7,14)(8,15,16,20)(11,18,13,19)(21,25)(22,29)(23,33)(24,27)(26,36)(28,34)(30,32)(31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7847135);