#7647122

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6520 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)(5,6,10,13)(7,14,8,11)(9,17,18,12)(15,19,16,20)(22,26)(23,29)(24,32)(34,36);(1,2)(3,4)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(5,7,10,8)(6,11,13,14)(9,15,18,16)(12,19,17,20)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,6)(7,11)(8,14)(9,12)(10,13)(15,19)(16,20)(17,18)(21,24,25,32)(22,28,26,33)(23,31,29,35)(27,34,30,36);(5,8,10,7)(6,11,13,14)(9,16,18,15)(12,19,17,20);(5,9)(6,12)(7,15)(8,16)(10,18)(11,19)(13,17)(14,20)

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=i,e^2=hi,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, 7554318);