#7641786

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3455	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6690 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)(5,6)(7,10)(8,16)(9,15)(11,13)(12,18)(14,17)(19,20)(21,22)(23,29)(24,32)(25,27)(26,30)(28,33)(31,34)(35,36);(2,4)(5,7)(6,9)(8,14)(10,13)(11,15)(12,17)(16,19)(18,20)(21,23)(22,26)(24,31)(25,30)(27,29)(28,34)(32,35)(33,36);(5,8)(6,12)(7,14)(9,17)(10,19)(11,18)(13,16)(15,20)(22,27)(23,30)(28,32)(31,36);(5,9,11,10)(6,7,13,15)(8,17,18,19)(12,14,16,20)(21,24,25,33)(22,28,27,32)(23,31,30,36)(26,34,29,35);(8,18)(12,16)(14,20)(17,19);(5,10,11,9)(6,7,13,15)(8,19,18,17)(12,14,16,20)

Polycyclic presentation:

2^9|a^2=gh,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi

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, 7548982);