#116863

order	512
nilpotency class	3
isoclinism	isoclinic class of #10320
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

820 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239162, #72240726, #72240729
maximal normal subgroup of	#72239162, #72240726, #72240729

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24059);