#7627668

order	512
nilpotency class	2
isoclinism	isoclinic class of #62972
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×#538	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6234 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)(5,6,10,8)(7,13,14,11)(9,12,17,16)(15,19,20,18)(21,22,27,29)(23,32,33,28)(24,34,31,26)(25,30,35,36);(7,14)(9,17)(11,13)(12,16)(21,23,27,33)(22,28,29,32)(24,35,31,25)(26,36,34,30);(6,8)(11,13)(12,16)(18,19)(22,29)(24,31)(25,35)(28,32);(5,7,10,14)(6,11,8,13)(9,15,17,20)(12,18,16,19)(21,24,27,31)(22,26,29,34)(23,25,33,35)(28,30,32,36);(5,8,10,6)(7,13,14,11)(9,16,17,12)(15,19,20,18)(21,25)(22,30)(23,31)(24,33)(26,32)(27,35)(28,34)(29,36);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,26)(22,31)(23,30)(24,29)(25,32)(27,34)(28,35)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534864);