#7645103

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

7156 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)(5,6,10,14)(7,12,8,11)(9,13,17,20)(15,19,16,18)(24,32)(28,35)(30,36)(33,34);(1,2)(3,4)(5,7,10,8)(6,11,14,12)(9,15,17,16)(13,18,20,19)(21,22)(23,27)(24,28)(25,29)(26,31)(30,34)(32,35)(33,36);(5,7,10,8)(6,11,14,12)(9,15,17,16)(13,18,20,19)(21,23)(22,26)(24,30)(25,31)(27,29)(28,33)(32,36)(34,35);(5,8,10,7)(6,12,14,11)(9,15,17,16)(13,18,20,19)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,9,10,17)(6,13,14,20)(7,15,8,16)(11,18,12,19)(21,22)(23,26)(24,28)(25,29)(27,31)(30,33)(32,35)(34,36);(5,6)(7,11)(8,12)(9,13)(10,14)(15,18)(16,19)(17,20)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=h,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, 7552299);