#7503744

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213801, #72216212, #72216384
maximal normal subgroup of	#72213801, #72216212, #72216384
maximal quotient of	#72213456

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410940);