#484079

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#509	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

466 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=gi,f^2=h,g^2=i,b^a=be,d^a=dh,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260491, #72320258, #72321792
maximal normal subgroup of	#72260491, #72320258, #72321792

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391275);