#6497740

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470901
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6404936);