#7732088

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

6465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7639284);