#479976

order	512
nilpotency class	2
isoclinism	isoclinic class of #180872
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂×#41	C₄²
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

457 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=g,f^2=h,h^2=i,b^a=bh,d^a=dg,f^a=fi,c^b=cg,e^b=ei,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286755, #72286841, #72286904, …
maximal normal subgroup of	#72286755, #72286841, #72286904, …
maximal quotient of	#72343334, #72343378, #72343410, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387172);