#6467452

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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
prime radical	#25779	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2686 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)(7,10)(8,9)(11,12)(13,14,17,21,18,22,25,19)(15,23,24,27,16,20,26,28);(2,4)(14,19)(15,24)(16,26)(17,25)(21,22)(27,28);(6,9)(10,12)(13,15,25,26,18,16,17,24)(14,20,19,27,22,23,21,28);(13,16,25,24,18,15,17,26)(14,20,19,27,22,23,21,28);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=hi,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319070, #72319085, #72319101, …
maximal normal subgroup of	#72319070, #72319085, #72319101, …
maximal quotient of	#72322975

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374648);