#7517135

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
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 ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2168 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)(7,12)(8,10)(9,11)(13,14,17,21)(15,22,25,19)(16,26,18,20)(23,28,24,27);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,24)(17,25)(18,23)(20,27)(21,22)(26,28);(2,4);(2,4)(5,8,9,7)(6,10,11,12)(15,23)(16,18)(19,27)(20,26)(22,28)(24,25);(6,11)(10,12)(13,16,17,18)(14,20,21,26)(15,24,25,23)(19,27,22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257481, #72257505, #72263854, …
maximal normal subgroup of	#72257481, #72257505, #72263854, …
maximal quotient of	#72307946

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7424331);