#194765

order	512
nilpotency class	3
isoclinism	isoclinic class of #18192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101961);