#7808372

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

6746 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7715568);