#181434

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
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	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1768 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)(15,23)(16,20)(17,21)(18,22)(19,24)(25,27)(26,28);(2,4)(13,15,17,26,18,25,16,24)(14,19,20,27,22,28,21,23);(6,9)(10,12)(13,16,18,17)(14,20,22,21)(15,24,25,26)(19,27,28,23);(15,25)(19,28)(23,27)(24,26)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310318, #72310321, #72310323
maximal normal subgroup of	#72310318, #72310321, #72310323

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88630);