#230225

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

641 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dhi,f^a=fi,d^b=de,e^b=eh,g^b=gh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137421);