#230316

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

688 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=ei,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,d^c=dh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137512);