#181483

order	512
nilpotency class	3
isoclinism	isoclinic class of #17392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1578 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)(3,11,14,7)(4,8)(5,9)(6,10)(12,16,13,15)(17,18,20,25,22,24,21,26)(19,23,28,31,27,30,29,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,16)(10,11)(17,19,21,29,22,27,20,28)(18,23,26,32,24,30,25,31);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,11,16)(19,27)(23,30)(28,29)(31,32);(1,5,6,4)(2,9,10,8)(3,13,14,12)(7,16,11,15)(18,24)(23,30)(25,26)(31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,f^2=i,g^2=i,b^a=be,d^a=di,e^a=eh,c^b=ci,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316215, #72316220, #72316222, …
maximal normal subgroup of	#72316215, #72316220, #72316222, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88679);