#136878

order	512
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
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

531 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225208, D₈×#357, #72226899
maximal normal subgroup of	#72225208, D₈×#357, #72226899
maximal quotient of	#72225188, #72226867, #72226887, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44074);