#484569

order	512
nilpotency class	4
isoclinism	isoclinic class of #12298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1613
prime radical	#60741	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

864 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=i,e^2=g,g^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gi,h^a=hi,e^b=egi,f^b=fhi,g^b=gi,e^c=eh,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306773, #72307091, #72307468
maximal normal subgroup of	#72306773, #72307091, #72307468

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391765);