#205199

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1060 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)(8,12)(9,14)(10,15)(11,16)(17,20)(18,19)(21,22,25,29,26,30,24,28)(23,31,32,27,34,35,33,36);(1,2,3,4)(5,7,9,8,11,18,10,17)(6,12,15,13,16,20,14,19)(21,23)(22,27)(24,32)(25,33)(26,34)(28,35)(29,31)(30,36);(5,8,9,18,11,17,10,7)(6,13,15,20,16,19,14,12);(5,9,11,10)(6,14,16,15)(7,8,18,17)(12,19,20,13)(21,24,26,25)(22,28,30,29)(23,32,34,33)(27,35,36,31)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ch,e^b=ei,g^b=gi

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, 112395);