#192070

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1072 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=hi,e^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,f^b=fi,e^c=ei

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, 99266);