#226230

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂²×C₄⋊C₄
prime radical	C₂²×D₄	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

714 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ef,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343127, #72343129, #72343177
maximal normal subgroup of	#72343127, #72343129, #72343177
maximal quotient of	#72343568, #72343581

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 133426);