#475907

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1032 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214428, #72216148, #72216714
maximal normal subgroup of	#72214428, #72216148, #72216714
maximal quotient of	#72214149

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383103);