#6371299

order	512
nilpotency class	3
isoclinism	isoclinic class of #60388
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

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×#585	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4654 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#60388, #72312195, #72312202
maximal normal subgroup of	C₄×#60388, #72312195, #72312202

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278495);