#481410

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	#61431	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

751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=f,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, #72260377, #72260397
maximal normal subgroup of	#72260372, #72260377, #72260397

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388606);