#394619

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264900, #72264924, #72264952
maximal normal subgroup of	#72264900, #72264924, #72264952

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301815);