#6374391

order	512
nilpotency class	3
isoclinism	isoclinic class of #60750
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,34)(35,36);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23,25,31)(22,26,28,34)(24,30,32,36)(27,33,29,35);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,27)(24,32)(26,33)(28,29)(30,36)(34,35);(6,12)(10,13)(11,20)(18,19);(22,28)(26,34)(27,29)(33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72292832, #72292836, #72292848
maximal normal subgroup of	#72292832, #72292836, #72292848

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281587);