#482158

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#2617	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

947 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,e^2=h,b^a=be,c^a=cgh,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284792, #72285719, #72285778
maximal normal subgroup of	#72284792, #72285719, #72285778

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389354);