#192274

order	512
nilpotency class	3
isoclinism	isoclinic class of #18252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

971 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,6)(3,4)(7,8)(9,10,14,18,15,19,13,28)(11,20,23,16,24,30,22,31)(12,17,25,33,27,34,26,32)(21,35,36,29,37,39,38,40);(1,2)(3,6)(4,8)(5,7)(9,11,14,23,15,24,13,22)(10,16,18,30,19,31,28,20)(12,21,25,36,27,37,26,38)(17,29,33,39,34,40,32,35);(3,4)(6,8)(9,12)(10,17)(11,21)(13,26)(14,25)(15,27)(16,29)(18,33)(19,34)(20,35)(22,38)(23,36)(24,37)(28,32)(30,39)(31,40);(1,3,5,4)(2,6,7,8)(9,13,15,14)(10,18,19,28)(11,22,24,23)(12,25,27,26)(16,30,31,20)(17,32,34,33)(21,36,37,38)(29,35,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99470);