#7718424

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×#3543	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7248 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)(8,10)(9,16)(12,17)(13,14)(15,20)(18,19)(24,32)(27,34)(30,36)(33,35);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,18)(16,17)(19,20)(21,22,25,28)(23,29,31,26)(24,27,32,34)(30,35,36,33);(6,8)(7,13)(12,16)(15,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,7,10,13)(6,11,8,14)(9,15,17,19)(12,18,16,20);(5,8,10,6)(7,14,13,11)(9,16,17,12)(15,20,19,18)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,24)(22,27)(23,30)(25,32)(26,33)(28,34)(29,35)(31,36)

Polycyclic presentation:

2^9|b^2=hi,d^2=h,e^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

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, 7625620);