#7627372

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×#121	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6420 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)(5,6,11,10)(7,13,8,12)(9,14,17,18)(15,20,16,19)(21,22)(23,27)(24,29)(25,30)(26,31)(28,33)(32,35)(34,36);(5,7,11,8)(6,12,10,13)(9,15,17,16)(14,19,18,20)(21,23,26,33)(22,27,31,28)(24,32,34,30)(25,29,35,36);(1,2,4,3)(5,7,11,8)(6,12,10,13)(9,15,17,16)(14,19,18,20)(21,23)(22,28)(24,32)(25,29)(26,33)(27,31)(30,34)(35,36);(5,8,11,7)(6,13,10,12)(9,16,17,15)(14,20,18,19);(1,3,4,2)(5,9)(6,14)(7,15)(8,16)(10,18)(11,17)(12,19)(13,20)(21,24)(22,29)(23,32)(25,28)(26,34)(27,35)(30,33)(31,36);(5,10,11,6)(7,13,8,12)(9,14,17,18)(15,19,16,20)(21,25)(22,30)(23,29)(24,27)(26,35)(28,34)(31,32)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=g,e^2=gi,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, 7534568);