#379724

order	512
nilpotency class	3
isoclinism	isoclinic class of #195972
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3117 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 3117 conjugacy classes of subgroups — too many to draw. Only its 314 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313588, #72313592, #72313638, …
maximal normal subgroup of	#72313588, #72313592, #72313638, …
maximal quotient of	#72314152, #72314197

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286920);