#7627243

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3475	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6792 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)(7,12)(8,11)(9,13)(10,15)(14,18)(16,19)(17,20)(22,27)(23,29)(30,35)(31,36);(5,6)(7,12)(8,11)(9,13)(10,15)(14,18)(16,19)(17,20)(21,22)(23,28)(24,29)(25,30)(26,27)(31,33)(32,36)(34,35);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,22)(23,28)(24,29)(25,30)(26,27)(31,33)(32,36)(34,35);(5,8,9,16)(6,11,13,19)(7,14,15,20)(10,17,12,18)(21,23,26,29)(22,28,27,24)(25,31,34,36)(30,33,35,32);(5,7)(6,12)(8,14)(9,15)(10,13)(11,18)(16,20)(17,19)(21,24,26,28)(22,29,27,23)(25,33,34,32)(30,31,35,36);(5,6)(7,10)(8,11)(9,13)(12,15)(14,17)(16,19)(18,20)(21,25,26,34)(22,30,27,35)(23,31,29,36)(24,32,28,33)

Polycyclic presentation:

2^9|a^2=i,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534439);