#6345832

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61441	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2922 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1545 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,f^a=fh,f^b=fh,d^c=dh,e^d=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323047
maximal normal subgroup of	#72323047

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253028);