#7744643

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
prime radical	C₂×#3542	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6738 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)(6,10)(11,17)(12,15)(18,20)(21,22)(23,27)(24,31)(25,33)(26,29)(28,34)(30,36)(32,35);(1,2)(3,4)(5,6)(7,11)(8,15)(9,10)(12,16)(13,17)(14,20)(18,19)(21,23,26,24)(22,27,29,31)(25,30,34,32)(28,35,33,36);(6,10)(7,13)(12,15)(14,19)(21,24,26,23)(22,27,29,31)(25,32,34,30)(28,35,33,36);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,22)(23,27)(24,31)(25,28)(26,29)(30,35)(32,36)(33,34);(5,7,9,13)(6,11,10,17)(8,14,16,19)(12,18,15,20)(21,25)(22,28)(23,30)(24,32)(26,34)(27,35)(29,33)(31,36);(5,8,9,16)(6,12,10,15)(7,14,13,19)(11,18,17,20)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7651839);