#481304

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
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	#60695	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

758 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)(19,24)(20,23)(21,34)(22,27)(25,31)(26,32)(28,38)(29,35)(30,44)(33,37)(36,45)(39,43)(40,42)(41,46)(47,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(18,23)(20,31)(21,32)(24,35)(25,26)(27,37)(29,43)(30,42)(33,45)(34,36)(38,46)(39,40)(41,48)(44,47);(3,9)(6,12)(10,15)(13,16)(17,19)(18,24)(20,29)(21,30)(22,28)(23,35)(25,39)(26,40)(27,38)(31,43)(32,42)(33,41)(34,44)(36,47)(37,46)(45,48);(19,28)(24,38)(29,41)(30,42)(35,46)(39,47)(40,44)(43,48)

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,c^b=ch,e^b=eg,g^b=gi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260361, #72260364, #72260397, …
maximal normal subgroup of	#72260361, #72260364, #72260397, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388500);