#137346

order	512
nilpotency class	4
isoclinism	isoclinic class of #12152
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

685 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 685 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=gi,d^2=gi,e^2=gi,f^2=gi,b^a=bd,c^a=ce,d^a=di,e^a=eghi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=egi,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227645
maximal normal subgroup of	#72227645

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44542);