#407019

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=efh,c^2=ef,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314215);