#7712594

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	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6814 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:

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

Polycyclic presentation:

2^9|a^2=g,b^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, 7619790);