#7715246

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

7130 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)(8,16)(12,17)(14,20)(18,19)(21,22,27,24)(23,28,30,26)(25,32,33,29)(31,34,36,35);(1,2)(3,4)(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,23,27,30)(22,28,24,26)(25,31,33,36)(29,35,32,34);(6,10)(11,13)(12,17)(18,19)(22,24)(23,30)(25,33)(34,35);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,24,27,22)(23,28,30,26)(25,32,33,29)(31,35,36,34);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,25)(22,29)(23,31)(24,32)(26,34)(27,33)(28,35)(30,36);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,26,27,28)(22,30,24,23)(25,34,33,35)(29,36,32,31)

Polycyclic presentation:

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

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, 7622442);