#204378

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1208 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ch,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215888, #72216051, #72216055
maximal normal subgroup of	#72215888, #72216051, #72216055

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111574);