#7712587

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×#3444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6776 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)(8,15)(12,17)(13,19)(18,20)(21,22,27,28)(23,25,32,29)(24,30,35,36)(26,33,34,31);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(22,28)(23,32)(30,36)(31,33);(6,10)(11,16)(12,17)(18,20)(21,23,27,32)(22,29,28,25)(24,31,35,33)(26,30,34,36);(5,7,9,14)(6,11,10,16)(8,13,15,19)(12,18,17,20)(21,24)(22,30)(23,33)(25,34)(26,29)(27,35)(28,36)(31,32);(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(21,25,27,29)(22,23,28,32)(24,34,35,26)(30,33,36,31);(5,8,9,15)(6,12,10,17)(7,13,14,19)(11,18,16,20)(21,26,27,34)(22,31,28,33)(23,30,32,36)(24,29,35,25)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619783);