#138642

order	512
nilpotency class	4
isoclinism	isoclinic class of #12430
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#466
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#1667	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

459 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=ghi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₈×#357, #72226892, #72226895
maximal normal subgroup of	D₈×#357, #72226892, #72226895
maximal quotient of	#72226866, #72226886, #72226914

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45838);