#7627082

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×#3566	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6564 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)(7,14)(8,11)(9,17)(10,13)(12,18)(15,19)(16,20)(22,27)(23,29)(25,33)(32,35);(1,2,4,3)(21,22)(23,28)(24,29)(25,30)(26,27)(31,35)(32,36)(33,34);(5,7,10,8)(6,11,13,14)(9,15,18,16)(12,19,17,20)(21,23)(22,24)(25,31)(26,29)(27,28)(30,32)(33,36)(34,35);(5,8,10,7)(6,11,13,14)(9,16,18,15)(12,19,17,20)(21,23)(22,28)(24,27)(25,31)(26,29)(30,35)(32,34)(33,36);(1,3,4,2)(5,6)(7,11)(8,14)(9,12)(10,13)(15,19)(16,20)(17,18)(21,24,26,28)(22,29,27,23)(25,32,33,35)(30,36,34,31);(5,9)(6,12)(7,15)(8,16)(10,18)(11,19)(13,17)(14,20)(21,25,26,33)(22,30,27,34)(23,31,29,36)(24,32,28,35)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=gh,e^2=gi,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, 7534278);