#115300

order	512
nilpotency class	3
isoclinism	isoclinic class of #100171
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#392	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1256 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 150 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 — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=dh,d^c=di,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285052, #72285120, #72285298
maximal normal subgroup of	#72285052, #72285120, #72285298

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22496);