#181472

order	512
nilpotency class	3
isoclinism	isoclinic class of #3192
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×C₂²⋊C₄
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1736 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)(5,6,10,11)(7,9,12,8)(13,14,17,20,18,21,16,26)(15,22,24,27,23,28,25,19);(2,4)(5,7,9,11,10,12,8,6)(13,15)(14,19,21,27)(16,25)(17,24)(18,23)(20,22,26,28);(15,23)(19,27)(22,28)(24,25);(5,8,10,9)(6,7,11,12)(13,16,18,17)(14,20,21,26)(15,24,23,25)(19,28,27,22)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238824
maximal normal subgroup of	#72238824

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88668);