#612382

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3490 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 3490 conjugacy classes of subgroups — too many to draw. Only its 610 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) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,e^2=fhi,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72266639
maximal subgroup of	#72266591, #72266639, #72267559
maximal normal subgroup of	#72266591, #72266639, #72267559
maximal quotient of	#72266629, #72267563

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 519578);