#7647056

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	#36721	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6700 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)(6,11)(7,12)(9,17)(15,20)(22,26)(23,29)(28,33)(30,35);(1,2)(3,4)(22,26)(27,32)(28,33)(34,36);(5,6,10,11)(7,8,12,16)(9,13,17,18)(14,15,19,20)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36);(5,7)(6,8)(9,14)(10,12)(11,16)(13,15)(17,19)(18,20)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,8)(6,12)(7,11)(9,15)(10,16)(13,19)(14,18)(17,20)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36);(5,9)(6,13)(7,14)(8,15)(10,17)(11,18)(12,19)(16,20)

Polycyclic presentation:

2^9|c^2=hi,d^2=i,e^2=i,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7554252);