#7711615

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	#37703	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6772 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)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,22)(23,25)(24,32)(26,30)(27,29)(28,34)(31,35)(33,36);(1,2)(3,4)(21,23,26,29)(22,27,30,25)(24,31,34,36)(28,35,32,33);(6,10)(11,13)(12,17)(18,19)(21,22,26,30)(23,25,29,27)(24,32,34,28)(31,35,36,33);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,35)(32,34)(33,36);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,24)(22,28)(23,31)(25,33)(26,34)(27,35)(29,36)(30,32);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,25,26,27)(22,29,30,23)(24,33,34,35)(28,36,32,31)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=hi,f^2=h,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, 7618811);