#154084

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂³	C₂²×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

3666 conjugacy classes of subgroups; only the 1106 normal subgroups are shown.

normal subgroups only

This group has 3666 conjugacy classes of subgroups — too many to draw. Only its 1106 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fg,d^2=f,b^a=be,c^a=cf,d^a=dg

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237326, #72237327
maximal normal subgroup of	#72237326, #72237327

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61280);