#185385

order	512
nilpotency class	3
isoclinism	isoclinic class of #185249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3497	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3239 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:

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

Polycyclic presentation:

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

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, 92581);