#6371441

order	512
nilpotency class	3
isoclinism	isoclinic class of #60388
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3585	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6217 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,f^b=fi,h^b=hi,g^c=gi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#60388, #72312221, #72312238
maximal normal subgroup of	C₄×#60388, #72312221, #72312238

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278637);