#7657741

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7038 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)(8,16)(11,18)(14,20)(17,19)(21,22,26,24)(23,29,31,27)(25,28,33,32)(30,35,36,34);(1,2,3,4)(5,6)(7,12)(8,11)(9,13)(10,15)(14,19)(16,18)(17,20)(21,22,26,24)(23,27,31,29)(25,28,33,32)(30,34,36,35);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(5,8,9,16)(6,11,13,18)(7,14,15,20)(10,17,12,19);(5,7)(6,12)(8,14)(9,15)(10,13)(11,19)(16,20)(17,18)(21,24,26,22)(23,29,31,27)(25,28,33,32)(30,34,36,35);(5,6)(7,10)(8,11)(9,13)(12,15)(14,17)(16,18)(19,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,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, 7564937);