#7711532

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6372 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)(5,6,11,8)(7,12,10,14)(9,16,18,13)(15,20,17,19)(24,32)(28,33)(30,36)(34,35);(1,2,3,4)(5,7,11,10)(6,12,8,14)(9,15,18,17)(13,19,16,20)(22,26)(27,29)(28,33)(34,35);(6,8)(12,14)(13,16)(19,20)(21,22,25,26)(23,29,31,27)(24,28,32,33)(30,35,36,34);(5,8,11,6)(7,14,10,12)(9,16,18,13)(15,20,17,19);(5,9,11,18)(6,13,8,16)(7,15,10,17)(12,19,14,20)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(5,10,11,7)(6,14,8,12)(9,17,18,15)(13,20,16,19)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=h,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, 7618728);