#7627076

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

6544 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)(22,28)(23,24)(26,30)(32,34);(1,2,4,3)(21,22,27,28)(23,31,24,29)(25,30,36,26)(32,33,34,35);(5,7,10,8)(6,11,13,14)(9,15,18,16)(12,19,17,20)(21,23,27,24)(22,29,28,31)(25,32,36,34)(26,33,30,35);(5,8,10,7)(6,11,13,14)(9,16,18,15)(12,19,17,20)(21,24,27,23)(22,29,28,31)(25,32,36,34)(26,35,30,33);(1,3,4,2)(5,6)(7,11)(8,14)(9,12)(10,13)(15,19)(16,20)(17,18)(21,25)(22,30)(23,32)(24,34)(26,28)(27,36)(29,35)(31,33);(5,9)(6,12)(7,15)(8,16)(10,18)(11,19)(13,17)(14,20)(21,26,27,30)(22,25,28,36)(23,33,24,35)(29,32,31,34)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=h,e^2=i,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, 7534272);