#6524772

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500044
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3517	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3154 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6431968);