#7145807

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2427 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)(7,11)(10,12)(13,14,18,22,19,23,17,16)(15,24,25,27,26,21,28,20);(6,9)(7,11)(13,15)(14,20)(16,24)(17,25)(18,28)(19,26)(21,22)(23,27);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,27)(17,28)(18,25)(19,26)(20,22)(23,24);(6,9)(10,12)(13,16,18,14,19,22,17,23)(15,20,25,24,26,27,28,21);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,16,23,22)(15,25,26,28)(20,24,27,21)

Polycyclic presentation:

2^9|a^2=f,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283111, #72284255, #72284532
maximal normal subgroup of	#72283111, #72284255, #72284532

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7053003);