#128440

order	512
nilpotency class	3
isoclinism	isoclinic class of #105580
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1749
prime radical	C₂⁴	#97
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

340 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(2,7,9,8)(10,14,16,15)(17,18,21,22,23,26,20,25)(19,27,28,32,31,29,30,24);(1,2,4,10,6,7,13,14,5,9,12,16,3,8,11,15)(17,19,22,29,21,30,26,32,23,31,25,27,20,28,18,24);(2,8,9,7)(4,12)(10,14,16,15)(11,13)(17,18,21,22,23,26,20,25)(19,27,28,32,31,29,30,24)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=gi,f^2=h,h^2=i,b^a=bd,f^a=fgi,c^b=ch,d^b=dg,e^b=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285170, #72285281
maximal normal subgroup of	#72285170, #72285281
maximal quotient of	#72319527, #72319543, #72319824

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35636);