#7627537

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
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×#3514	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6826 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)(7,15)(8,18)(9,12)(10,14)(11,13)(16,17)(19,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,34)(33,35);(5,7,12,11)(6,13,9,15)(8,16,14,19)(10,17,18,20)(21,22)(23,30)(24,32)(25,28)(26,29)(27,34)(31,33)(35,36);(5,8)(6,10)(7,16)(9,18)(11,19)(12,14)(13,17)(15,20)(21,23,26,28)(22,25,29,30)(24,31,34,35)(27,33,32,36);(5,9,12,6)(7,13,11,15)(8,18,14,10)(16,17,19,20)(21,24)(22,27)(23,31)(25,33)(26,34)(28,35)(29,32)(30,36);(5,10,12,18)(6,14,9,8)(7,17,11,20)(13,19,15,16)(21,22)(23,25)(24,27)(26,29)(28,30)(31,33)(32,34)(35,36);(5,11,12,7)(6,13,9,15)(8,19,14,16)(10,17,18,20)(21,25)(22,28)(23,29)(24,33)(26,30)(27,35)(31,32)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534733);