#481414

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×#102
prime radical	#61436	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,20,26,23,28,22,25)(19,29,30,45,32,46,38,43)(21,34,33,44,37,42,36,27)(24,35,40,47,41,48,39,31);(17,19,23,32)(18,24,28,41)(20,30,22,38)(21,35,37,48)(25,39,26,40)(27,43,44,45)(29,42,46,34)(31,33,47,36);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,20,23,22)(18,25,28,26)(19,30,32,38)(21,33,37,36)(24,39,41,40)(27,42,44,34)(29,43,46,45)(31,35,47,48);(3,9)(6,12)(10,15)(13,16)(17,20,23,22)(18,26,28,25)(19,30,32,38)(21,33,37,36)(24,40,41,39)(27,34,44,42)(29,45,46,43)(31,35,47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,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	#72260372, #72260375, #72260397
maximal normal subgroup of	#72260372, #72260375, #72260397

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388610);