#7626772

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×#3520	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6820 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)(8,18)(9,12)(10,19)(11,13)(15,20)(16,17)(21,22,27,31)(23,30,26,28)(24,25,29,36)(32,35,34,33);(1,3,4,2)(5,7,11,9)(6,12,13,14)(8,16,19,15)(10,17,18,20);(5,8)(6,10)(7,15)(9,16)(11,19)(12,20)(13,18)(14,17)(21,23,27,26)(22,28,31,30)(24,32,29,34)(25,33,36,35);(5,8)(6,10)(7,16)(9,15)(11,19)(12,17)(13,18)(14,20)(21,24)(22,25)(23,32)(26,34)(27,29)(28,33)(30,35)(31,36);(5,9,11,7)(6,12,13,14)(8,16,19,15)(10,20,18,17)(21,25)(22,29)(23,33)(24,31)(26,35)(27,36)(28,34)(30,32);(5,10)(6,8)(7,17)(9,20)(11,18)(12,16)(13,19)(14,15)(21,26,27,23)(22,30,31,28)(24,34,29,32)(25,35,36,33)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,e^2=g,f^2=gh,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, 7533968);