#7627558

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×#3546	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7040 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=h,e^2=i,f^2=gh,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, 7534754);