#7627184

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×#3565	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6444 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)(5,6,10,13)(7,14,8,11)(9,17,18,12)(15,19,16,20)(21,22,27,25)(23,28,34,33)(24,30,35,29)(26,36,32,31);(1,2,4,3)(5,7,10,8)(6,11,13,14)(9,15,18,16)(12,19,17,20)(21,23,27,34)(22,28,25,33)(24,32,35,26)(29,36,30,31);(5,6)(7,14)(8,11)(9,12)(10,13)(15,20)(16,19)(17,18)(21,24)(22,29)(23,32)(25,30)(26,34)(27,35)(28,36)(31,33);(5,6)(7,11)(8,14)(9,12)(10,13)(15,19)(16,20)(17,18)(21,25,27,22)(23,33,34,28)(24,30,35,29)(26,36,32,31);(1,3,4,2)(5,8,10,7)(6,11,13,14)(9,16,18,15)(12,19,17,20)(21,24)(22,30)(23,32)(25,29)(26,34)(27,35)(28,31)(33,36);(5,9,10,18)(6,12,13,17)(7,15,8,16)(11,19,14,20)(21,26,27,32)(22,31,25,36)(23,24,34,35)(28,29,33,30)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gh,e^2=ghi,f^2=h,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, 7534380);