#354969

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1378 conjugacy classes of subgroups; only the 626 normal subgroups are shown.

normal subgroups only

This group has 1378 conjugacy classes of subgroups — too many to draw. Only its 626 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=eh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257825, #72257838, #72257848
maximal normal subgroup of	#72257825, #72257838, #72257848

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262165);