#354022

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
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

3298 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315682, #72315697, #72315738, …
maximal normal subgroup of	#72315682, #72315697, #72315738, …
maximal quotient of	#72321576, #72321778

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261218);