#7522930

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
prime radical	#2433	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2062 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)(5,6,9,12)(7,11,8,10)(13,14,16,19)(15,21,23,18)(17,20,25,28)(22,27,24,26);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,18)(16,23)(17,22)(19,21)(20,26)(24,25)(27,28);(2,4);(2,4)(5,7,9,8)(6,11,12,10)(15,22)(17,25)(18,26)(20,28)(21,27)(23,24);(5,8,9,7)(6,11,12,10)(14,19)(15,23)(20,28)(22,24)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308466, #72308888
maximal normal subgroup of	#72308466, #72308888

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7430126);