#192864

order	512
nilpotency class	3
isoclinism	isoclinic class of #18682
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

882 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:

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

Polycyclic presentation:

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

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, 100060);