#493469

order	512
nilpotency class	4
isoclinism	isoclinic class of #12727
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
socle	C₂	C₂×#1782
prime radical	#62815	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2209 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=ghi,f^c=fi,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261089, #72261192, #72303558, …
maximal normal subgroup of	#72261089, #72261192, #72303558, …
maximal quotient of	#72303564, #72303821

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400665);