#7626618

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	C₂⁴	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6344 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,e^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, 7533814);