#481402

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
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

752 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)(17,18,20,25,23,27,22,37)(19,28,29,45,31,46,38,42)(21,33,32,43,36,44,35,26)(24,34,39,47,41,48,40,30);(17,19,23,31)(18,24,27,41)(20,29,22,38)(21,34,36,48)(25,39,37,40)(26,42,43,45)(28,44,46,33)(30,32,47,35);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,20,23,22)(18,25,27,37)(19,29,31,38)(21,32,36,35)(24,39,41,40)(26,33,43,44)(28,45,46,42)(30,34,47,48);(3,9)(6,12)(10,15)(13,16)(17,20,23,22)(18,25,27,37)(19,29,31,38)(21,32,36,35)(24,39,41,40)(26,33,43,44)(28,45,46,42)(30,34,47,48)

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,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234338, #72260370, #72260397
maximal normal subgroup of	#72234338, #72260370, #72260397

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388598);