#7613952

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#28467	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|c^2=gi,d^2=hi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308141, #72308936
maximal normal subgroup of	#72308141, #72308936

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521148);