#354782

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3830 conjugacy classes of subgroups; only the 626 normal subgroups are shown.

normal subgroups only

This group has 3830 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,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	#72286571, #72307605, #72307616, …
maximal normal subgroup of	#72286571, #72307605, #72307616, …
maximal quotient of	#72286827

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261978);