#149036

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1934
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,3)(5,6,10,16,11,17,33,48,12,18,34,49,32,47,9,15)(7,19,22,39,23,46,21,50,24,52,31,13,35,37,57,38)(8,14,28,43,29,44,27,42,30,45,61,67,60,66,26,41)(20,51,54,64,55,68,53,65,56,40,62,36,59,63,25,58);(1,2,4,3)(5,7)(6,13)(8,25)(9,31)(10,21)(11,35)(12,24)(14,40)(15,46)(16,19)(17,50)(18,39)(20,27)(22,33)(23,32)(26,56)(28,55)(29,62)(30,53)(34,57)(36,42)(37,48)(38,47)(41,64)(43,58)(44,68)(45,51)(49,52)(54,60)(59,61)(63,66)(65,67);(1,3,4,2)(5,8,9,26,32,60,34,61,12,30,33,27,11,29,10,28)(6,14,15,41,47,66,49,67,18,45,48,42,17,44,16,43)(7,20,21,53,35,59,22,54,24,56,57,25,23,55,31,62)(13,36,19,51,50,65,37,63,39,64,52,40,38,58,46,68)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=hi,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fh,g^b=gh,h^b=hi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56232);