#412128

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#493	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2140 conjugacy classes of subgroups; only the 302 normal subgroups are shown.

normal subgroups only

This group has 2140 conjugacy classes of subgroups — too many to draw. Only its 302 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=ei,c^2=e,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265008, #72265027, #72265127
maximal normal subgroup of	#72265008, #72265027, #72265127

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319324);