#7627209

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×#540	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6448 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)(5,6,10,14)(7,15,17,11)(8,18,16,12)(9,13,19,20)(22,25)(27,30)(28,33)(35,36);(1,2,4,3)(5,7)(6,11)(8,19)(9,16)(10,17)(12,20)(13,18)(14,15);(5,8,10,16)(6,12,14,18)(7,9,17,19)(11,13,15,20)(21,22,26,25)(23,27,31,30)(24,28,34,33)(29,35,32,36);(5,7)(6,11)(8,9)(10,17)(12,13)(14,15)(16,19)(18,20)(21,23,26,31)(22,27,25,30)(24,32,34,29)(28,36,33,35);(1,3,4,2)(5,9)(6,13)(7,16)(8,17)(10,19)(11,18)(12,15)(14,20)(21,24,26,34)(22,28,25,33)(23,29,31,32)(27,35,30,36);(5,6)(7,11)(8,12)(9,13)(10,14)(15,17)(16,18)(19,20)(21,25,26,22)(23,30,31,27)(24,33,34,28)(29,36,32,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=gh,e^2=hi,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, 7534405);