#417895

order	512
nilpotency class	3
isoclinism	isoclinic class of #417496
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	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2151 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 2151 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) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14)(15,19)(16,20)(17,21)(18,22)(23,33)(24,31)(25,32)(26,34)(27,35)(28,36)(29,37)(30,39)(38,43)(40,42)(41,44);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,23)(17,24)(18,25)(20,33)(21,31)(22,32)(26,38)(27,39)(28,40)(29,41)(30,35)(34,43)(36,42)(37,44);(13,16,18,27)(14,20,22,35)(15,23,25,39)(17,28,29,26)(19,30,32,33)(21,36,37,34)(24,40,41,38)(31,43,44,42);(2,4)(7,12)(9,11)(13,17,18,29)(14,21,22,37)(15,24,25,41)(16,26,27,28)(19,31,32,44)(20,34,35,36)(23,38,39,40)(30,42,33,43)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268304
maximal normal subgroup of	#72268304
maximal quotient of	#72268317

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325091);