#7865057

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	#34732	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6547 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)(7,8)(9,13)(10,11)(12,16)(14,15)(17,18)(19,20)(21,22)(23,27)(24,25)(26,34)(28,30)(29,35)(31,32)(33,36);(1,2)(3,4)(21,23,28,25)(22,27,30,24)(26,31,35,33)(29,36,34,32);(6,11)(7,12)(13,18)(14,19)(21,24,28,27)(22,23,30,25)(26,32,35,36)(29,31,34,33);(5,7,10,12)(6,8,11,16)(9,14,17,19)(13,15,18,20)(21,25,28,23)(22,24,30,27)(26,33,35,31)(29,32,34,36);(5,8)(6,12)(7,11)(9,15)(10,16)(13,19)(14,18)(17,20)(21,26)(22,29)(23,31)(24,32)(25,33)(27,36)(28,35)(30,34);(5,9,10,17)(6,13,11,18)(7,14,12,19)(8,15,16,20)(21,27,28,24)(22,23,30,25)(26,32,35,36)(29,33,34,31)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7772253);