#7627132

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×#121	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6356 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)(5,6,9,13)(7,11,15,10)(8,16,17,12)(14,18,20,19)(21,22,27,25)(23,30,26,28)(24,29,33,34)(31,36,32,35);(1,2,4,3)(5,7)(6,10)(8,14)(9,15)(11,13)(12,18)(16,19)(17,20);(5,6)(7,11)(8,12)(9,13)(10,15)(14,19)(16,17)(18,20)(21,23,27,26)(22,28,25,30)(24,31,33,32)(29,35,34,36);(5,6)(7,10)(8,12)(9,13)(11,15)(14,18)(16,17)(19,20)(21,24)(22,29)(23,31)(25,34)(26,32)(27,33)(28,35)(30,36);(1,3,4,2)(5,7)(6,11)(8,14)(9,15)(10,13)(12,19)(16,18)(17,20)(21,25,27,22)(23,30,26,28)(24,29,33,34)(31,35,32,36);(5,8,9,17)(6,12,13,16)(7,14,15,20)(10,18,11,19)(21,26,27,23)(22,30,25,28)(24,32,33,31)(29,36,34,35)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,e^2=gi,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, 7534328);