#7739313

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	#37302	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6686 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)(21,22,28,30)(23,24,26,25)(27,29,35,36)(31,32,34,33);(1,2)(3,4)(5,6)(7,11)(8,15)(9,10)(12,16)(13,17)(14,20)(18,19)(21,23,28,26)(22,25,30,24)(27,34,35,31)(29,32,36,33);(6,10)(7,13)(12,15)(14,19)(21,24,28,25)(22,23,30,26)(27,32,35,33)(29,31,36,34);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,25,28,24)(22,23,30,26)(27,33,35,32)(29,31,36,34);(5,7,9,13)(6,11,10,17)(8,14,16,19)(12,18,15,20)(21,26,28,23)(22,24,30,25)(27,34,35,31)(29,32,36,33);(5,8,9,16)(6,12,10,15)(7,14,13,19)(11,18,17,20)(21,27,28,35)(22,29,30,36)(23,31,26,34)(24,32,25,33)

Polycyclic presentation:

2^9|a^2=g,b^2=h,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=dhi,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, 7646509);