#354873

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317237, #72317292, #72317303
maximal normal subgroup of	#72317237, #72317292, #72317303

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262069);