#6370735

order	512
nilpotency class	3
isoclinism	isoclinic class of #60399
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹²

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×H(7,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

17537 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)(3,4)(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19);(2,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,17)(16,18)(19,20);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20);(8,16)(11,18)(14,20)(17,19)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72341287, #72341289, #72341350, …
maximal subgroup of	#72227973, #72227981, #72227984, …
maximal normal subgroup of	#72227973, #72227981, #72233689, …
maximal quotient of	#72232917, #72233637, #72233659, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277931);