#7504014

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500661
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 ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2038 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)(7,8)(9,11)(10,12)(13,14,18,19,17,22,15,21)(16,24,23,28,25,27,26,20);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,18)(14,19,22,21)(16,23,25,26)(20,27,28,24);(2,4)(13,16,17,25)(14,20,22,28)(15,23,18,26)(19,27,21,24);(6,11)(7,12)(14,21)(15,18)(19,22)(20,24)(23,26)(27,28);(5,8,9,10)(6,7,11,12)(14,22)(19,21)(20,28)(24,27)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307757, #72307964, #72308482
maximal normal subgroup of	#72307757, #72307964, #72308482

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411210);