#205615

order	512
nilpotency class	3
isoclinism	isoclinic class of #205565
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60700	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1011 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305445, #72305449, #72305488
maximal normal subgroup of	#72305445, #72305449, #72305488

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112811);