#227576

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

736 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)(5,6,9,10)(7,11,12,8)(13,14,17,19)(15,22,24,28)(16,21,18,25)(20,26,27,23);(1,3,4,2)(6,7)(8,11)(10,12)(14,19)(20,27)(21,25)(22,28);(6,10)(7,12)(13,15,17,24)(14,20,19,27)(16,23,18,26)(21,28,25,22);(5,7)(6,11)(8,10)(9,12)(13,16,17,18)(14,21,19,25)(15,23,24,26)(20,28,27,22)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dei,e^b=eh,d^c=dh,f^c=fi,e^d=eh

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, 134772);