#383757

order	512
nilpotency class	3
isoclinism	isoclinic class of #213760
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#35358	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1662 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251367, #72251392, #72251399
maximal normal subgroup of	#72251367, #72251392, #72251399
maximal quotient of	#72251455

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290953);