#417884

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2194 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 2194 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=dei,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263014, #72263056
maximal normal subgroup of	#72263014, #72263056

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325080);