#160416

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

2340 conjugacy classes of subgroups; only the 530 normal subgroups are shown.

normal subgroups only

This group has 2340 conjugacy classes of subgroups — too many to draw. Only its 530 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) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=efgh,c^2=fg,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72266659, #72266662
Fitting subgroup of	#72266659, #72266660, #72266661
maximal subgroup of	#72266620, #72266623, #72266629, …
maximal normal subgroup of	#72266620, #72266623, #72266629, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 67612);