#7627332

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	C₂⁴	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=g,e^2=h,f^2=h,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, 7534528);