#148989

order	512
nilpotency class	4
isoclinism	isoclinic class of #13398
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#481
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1853
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(5,6,11,18,13,19,12,32)(7,20,24,36,23,35,8,25)(9,27,21,17,10,30,29,16)(14,31,34,28,33,26,15,22);(1,2)(3,4)(5,7,10,22,11,24,29,31,13,23,9,28,12,8,21,26)(6,14,17,20,32,15,27,25,19,33,16,35,18,34,30,36);(2,4)(5,8,9,26,11,7,21,22,13,24,10,31,12,23,29,28)(6,15,16,25,32,33,30,35,19,34,17,36,18,14,27,20)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215378, #72216650, #72217040
maximal normal subgroup of	#72215378, #72216650, #72217040
maximal quotient of	#72215302

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56185);