#323127

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
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	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

801 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)(6,10)(7,12)(8,11)(13,14,18,26,17,22,16,21)(15,19,23,20,25,28,24,27);(5,6,11,12,9,7,8,10)(13,15,18,23,17,25,16,24)(14,19,26,20,22,28,21,27);(2,4)(13,15,17,25)(14,20,22,27)(16,23,18,24)(19,21,28,26);(5,7,8,12,9,6,11,10)(13,16,17,18)(14,21,22,26)(15,23,25,24)(19,20,28,27)

Polycyclic presentation:

2^9|a^2=fi,b^2=eghi,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230323);