#7995383

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

6373 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)(7,11)(8,15)(9,10)(12,16)(13,17)(14,20)(18,19)(21,22,25,27)(23,26,29,34)(24,28,32,35)(30,33,31,36);(1,2,3,4)(7,13)(11,17)(14,19)(18,20)(21,23)(22,26)(24,31)(25,29)(27,34)(28,36)(30,32)(33,35);(6,10)(11,17)(12,15)(18,20)(22,27)(23,29)(28,35)(30,31);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,23)(22,26)(24,31)(25,29)(27,34)(28,36)(30,32)(33,35);(5,7,9,13)(6,11,10,17)(8,14,16,19)(12,18,15,20)(21,24,25,32)(22,28,27,35)(23,30,29,31)(26,33,34,36);(5,8)(6,12)(7,14)(9,16)(10,15)(11,18)(13,19)(17,20)(21,22)(23,26)(24,28)(25,27)(29,34)(30,33)(31,36)(32,35)

Polycyclic presentation:

2^9|a^2=i,b^2=g,e^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=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, 7902579);