#124008

order	512
nilpotency class	3
isoclinism	isoclinic class of #2032
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1446
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303562, #72303837
maximal normal subgroup of	#72303562, #72303837
maximal quotient of	#72303588, #72304146

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31204);