#495759

order	512
nilpotency class	4
isoclinism	isoclinic class of #33568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#2202	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2,4,3)(5,6,10,15)(7,13,8,11)(9,12,18,14)(16,20,17,19);(6,11)(7,8)(9,18)(12,19)(13,15)(14,20);(6,12)(9,18)(11,19)(13,20)(14,15)(16,17);(1,3,4,2)(5,7,10,8)(6,13,15,11)(9,16,18,17)(12,20,14,19)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268162, C₄×#33570, #72310335
maximal normal subgroup of	#72268162, C₄×#33570
maximal quotient of	#72268164, #72282555, #72282732

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402955);