#139853

order	512
nilpotency class	4
isoclinism	isoclinic class of #139718
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

667 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 667 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=hi,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dgi,e^b=eh,f^b=fgi,g^b=gi,d^c=dh,f^c=fh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268489, #72268494, #72268496
maximal normal subgroup of	#72268489, #72268494, #72268496

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47049);