#7715428

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

7036 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)(8,16)(12,17)(14,20)(18,19)(21,22,28,32)(23,26,25,29)(24,31,27,30)(33,35,34,36);(1,2,3,4)(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,23)(22,29)(24,34)(25,28)(26,32)(27,33)(30,36)(31,35);(6,10)(11,13)(12,17)(18,19)(21,24,28,27)(22,30,32,31)(23,33,25,34)(26,36,29,35);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,25,28,23)(22,29,32,26)(24,33,27,34)(30,36,31,35);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,26,28,29)(22,23,32,25)(24,35,27,36)(30,34,31,33);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,27,28,24)(22,31,32,30)(23,33,25,34)(26,36,29,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622624);