#230485

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=eh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=chi,d^b=dei,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343506, #72343877, #72343884
maximal normal subgroup of	#72343506, #72343877, #72343884

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137681);