#183283

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

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

1082 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)(13,14,18,21,17,20,26,16)(15,22,25,23,24,19,27,28);(2,4)(5,6,7,9,8,10,11,12)(13,15)(14,19)(16,23)(17,24)(18,25)(20,22)(21,28)(26,27);(13,16,26,20,17,21,18,14)(15,23,27,22,24,28,25,19);(15,24)(19,22)(23,28)(25,27)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,f^2=h,g^2=i,b^a=be,d^b=dh

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224770, #72224775, #72224776, …
maximal normal subgroup of	#72224770, #72224775, #72224776, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90479);