#182503

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1166 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)(5,6,10,11)(7,9,12,8)(13,14,17,21,18,16,20,25)(15,19,24,27,23,22,26,28);(2,4)(5,7,9,11,10,12,8,6)(13,15)(14,19)(16,22)(17,24)(18,23)(20,26)(21,27)(25,28);(5,8,10,9)(6,7,11,12)(13,16,17,25,18,14,20,21)(15,22,24,28,23,19,26,27);(5,8,10,9)(6,7,11,12)(15,23)(19,22)(24,26)(27,28)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,e^a=eh,g^a=gh,c^b=ch,d^b=dhi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286059, #72343870, #72343875
maximal normal subgroup of	#72286059, #72343870, #72343875

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89699);