#204563

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^b=eh,g^b=gh,d^c=dhi

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, 111759);