#354780

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₈	C₂³
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂⁴	C₂³×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3942 conjugacy classes of subgroups; only the 642 normal subgroups are shown.

normal subgroups only

This group has 3942 conjugacy classes of subgroups — too many to draw. Only its 642 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)(13,14,17,21,19,22,18,16)(15,20,25,28,23,27,26,24);(2,4)(13,15,18,26,19,23,17,25)(14,20,16,24,22,27,21,28);(6,8)(15,23)(20,27)(24,28)(25,26);(13,16,18,22,19,21,17,14)(15,24,26,27,23,28,25,20)

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,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286545, #72308300, #72308309, …
maximal normal subgroup of	#72286545, #72308300, #72308309, …
maximal quotient of	#72286617, #72286827

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261976);