#500251

order	512
nilpotency class	4
isoclinism	isoclinic class of #500250
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24058
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3484	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

864 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,f^b=fh,d^c=dh,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^f=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407447);