#7626612

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
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×#3475	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6750 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)(5,6,12,16)(7,17,20,13)(8,9,10,14)(11,18,19,15)(23,29)(27,31)(30,35)(34,36);(5,7,12,20)(6,13,16,17)(8,15,10,18)(9,19,14,11)(21,22,26,25)(23,27,29,31)(24,28,32,33)(30,34,35,36);(5,8,12,10)(6,14,16,9)(7,18,20,15)(11,13,19,17)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,35)(33,36);(5,9)(6,8)(7,19)(10,16)(11,20)(12,14)(13,15)(17,18)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36);(5,10,12,8)(6,14,16,9)(7,18,20,15)(11,17,19,13)(21,25,26,22)(23,31,29,27)(24,28,32,33)(30,34,35,36);(5,11,12,19)(6,15,16,18)(7,9,20,14)(8,13,10,17)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,35)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533808);