#133124

order	512
nilpotency class	4
isoclinism	isoclinic class of #133077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

946 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 946 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,13,15,27)(4,16,18,9)(5,10,21,23)(6,12)(7,11)(8,14,25,31)(17,24,19,26)(20,28)(22,29,32,30);(1,3)(2,8,12,26)(4,6)(5,19)(7,15)(9,10,27,23)(11,24,28,25)(13,30,16,29)(14,22)(17,32)(18,20)(21,31);(1,4,6,3)(2,9,12,27)(5,17,21,14)(7,18,20,15)(8,23,26,10)(11,16,28,13)(19,22,31,32)(24,30,25,29)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293036
maximal normal subgroup of	#72293036
maximal quotient of	#72293156

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40320);