#7920208

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

6706 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)(5,6,11,12)(7,13,8,10)(9,17,19,14)(15,20,16,18)(21,22,26,30)(23,27,33,36)(24,28,32,35)(25,29,31,34);(1,2)(3,4)(5,7,11,8)(6,10,12,13)(9,15,19,16)(14,20,17,18)(21,23)(22,27)(24,31)(25,32)(26,33)(28,34)(29,35)(30,36);(6,12)(7,8)(14,17)(15,16)(21,24)(22,28)(23,25)(26,32)(27,29)(30,35)(31,33)(34,36);(5,8,11,7)(6,13,12,10)(9,16,19,15)(14,18,17,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,34)(32,35)(33,36);(5,9,11,19)(6,14,12,17)(7,15,8,16)(10,20,13,18)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,36)(30,35);(5,10,11,13)(6,7,12,8)(9,18,19,20)(14,16,17,15)(21,25,26,31)(22,29,30,34)(23,32,33,24)(27,35,36,28)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=h,e^2=h,f^2=i,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, 7827404);