#204551

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	#34977	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1222 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:

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

Polycyclic presentation:

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

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, 111747);