#226644

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

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×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

647 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,d^b=dei,e^b=eh,d^c=dhi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343763, #72343868, #72343878
maximal normal subgroup of	#72343763, #72343868, #72343878

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 133840);