#481399

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	#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

763 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)(17,18,21,26,23,27,22,37)(19,28,30,44,31,45,38,42)(20,32,34,43,36,47,35,25)(24,33,39,46,41,48,40,29);(17,19)(18,24)(20,33)(21,30)(22,38)(23,31)(25,42)(26,39)(27,41)(28,32)(29,35)(34,46)(36,48)(37,40)(43,44)(45,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16);(3,9)(6,12)(10,15)(13,16)

Polycyclic presentation:

2^9|a^2=f,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	#72234320, #72260370, #72260397, …
maximal normal subgroup of	#72234320, #72260370, #72260397, …
maximal quotient of	#72320236

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388595);