#7662435

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	#37278	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6543 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)(5,6,10,8)(7,11,13,14)(9,12,17,16)(15,18,19,20)(21,22)(23,31)(24,27)(25,29)(26,30)(28,33)(32,34)(35,36);(1,2)(3,4)(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19);(7,13)(11,14)(15,19)(18,20)(21,23,27,33)(22,28,24,31)(25,26,32,36)(29,35,34,30);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,24,27,22)(23,28,33,31)(25,34,32,29)(26,35,36,30);(5,8,10,6)(7,14,13,11)(9,12,17,16)(15,18,19,20)(21,25)(22,29)(23,26)(24,34)(27,32)(28,35)(30,31)(33,36);(5,9,10,17)(6,12,8,16)(7,15,13,19)(11,18,14,20)(21,26)(22,30)(23,32)(24,35)(25,33)(27,36)(28,29)(31,34)

Polycyclic presentation:

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

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, 7569631);