#6372733

order	512
nilpotency class	3
isoclinism	isoclinic class of #60399
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	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#3026	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4973 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)(10,16)(17,20);(5,6)(7,10)(8,11)(9,12)(13,17)(14,16)(15,18)(19,20);(5,7)(6,10)(8,13)(9,14)(11,17)(12,16)(15,19)(18,20);(1,3,4,2)(5,8)(6,11)(7,13)(9,15)(10,17)(12,18)(14,19)(16,20);(8,15)(11,18)(13,19)(17,20)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72341303, #72341309, #72341324
maximal subgroup of	#72240656, C₄×#60399, #72245877, …
maximal normal subgroup of	#72240656, C₄×#60399, #72245877, …
maximal quotient of	#72240674, #72267168, #72269812, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279929);