#484988

order	512
nilpotency class	4
isoclinism	isoclinic class of #12307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1616
prime radical	#60907	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

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

normal subgroups only

This group has 707 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=i,e^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,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, #72307084, #72307227, …
maximal normal subgroup of	#72306773, #72307084, #72307227, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392184);