#7648359

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

7032 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)(9,17)(12,19)(15,18)(16,20)(21,22)(23,28)(24,32)(25,26)(27,30)(29,35)(31,34)(33,36);(1,2)(3,4)(5,6)(7,14)(8,11)(9,12)(10,13)(15,20)(16,18)(17,19)(21,23,27,25)(22,28,30,26)(24,31,35,33)(29,36,32,34);(5,7,10,11)(6,8,13,14)(9,15,17,18)(12,16,19,20)(21,24)(22,29)(23,31)(25,33)(26,34)(27,35)(28,36)(30,32);(5,6)(7,8)(9,12)(10,13)(11,14)(15,16)(17,19)(18,20)(21,25,27,23)(22,28,30,26)(24,33,35,31)(29,36,32,34);(5,8)(6,11)(7,13)(9,16)(10,14)(12,18)(15,19)(17,20)(21,26,27,28)(22,25,30,23)(24,34,35,36)(29,33,32,31);(5,9,10,17)(6,12,13,19)(7,15,11,18)(8,16,14,20)(21,24)(22,29)(23,31)(25,33)(26,34)(27,35)(28,36)(30,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7555555);