#396721

order	512
nilpotency class	3
isoclinism	isoclinic class of #394463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#397
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1211 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252923, #72256905, #72256910
maximal normal subgroup of	#72252923, #72256905, #72256910

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303917);