#7627295

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×#3518	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6600 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)(5,6,12,10)(7,8,13,9)(11,14,19,18)(15,16,20,17)(21,22)(23,27)(24,29)(25,30)(26,31)(28,33)(32,35)(34,36);(5,7)(6,9)(8,10)(11,15)(12,13)(14,17)(16,18)(19,20)(21,23,26,33)(22,27,31,28)(24,32,34,30)(25,29,35,36);(1,3,4,2)(5,8,12,9)(6,7,10,13)(11,16,19,17)(14,15,18,20)(21,23)(22,28)(24,32)(25,29)(26,33)(27,31)(30,34)(35,36);(5,9,12,8)(6,13,10,7)(11,17,19,16)(14,20,18,15);(5,10,12,6)(7,8,13,9)(11,14,19,18)(15,17,20,16)(21,24)(22,29)(23,32)(25,28)(26,34)(27,35)(30,33)(31,36);(5,11,12,19)(6,14,10,18)(7,15,13,20)(8,16,9,17)(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=gh,c^2=i,d^2=g,e^2=g,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, 7534491);