#7518686

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2198 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:

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

Polycyclic presentation:

2^9|d^2=i,e^2=f,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^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307349, #72307490, #72307773
maximal normal subgroup of	#72307349, #72307490, #72307773

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7425882);