#7708291

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6718 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20);(2,4)(21,22)(23,29)(24,28)(25,27)(26,30)(31,35)(32,34)(33,36);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23)(22,26)(24,31)(25,30)(27,29)(28,33)(32,36)(34,35);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,27)(23,30)(28,34)(31,36);(6,12)(10,13)(11,20)(18,19)(23,30)(26,29)(31,36)(33,35);(21,24,25,32)(22,28,27,34)(23,31,30,36)(26,33,29,35)

Polycyclic presentation:

2^9|c^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei

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, 7615487);