#7565514

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2064 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)(8,12)(9,10)(15,23)(16,26)(17,25)(19,24)(20,27)(21,28);(1,2)(3,4)(6,10)(11,12)(13,14,18,22)(15,24,16,27)(17,28,25,21)(19,26,20,23);(5,7,9,8)(6,11,10,12)(13,15,17,23,18,16,25,26)(14,19,21,24,22,20,28,27);(2,4)(5,7,9,8)(6,11,10,12)(13,16,17,26,18,15,25,23)(14,20,21,27,22,19,28,24);(5,8,9,7)(6,11,10,12)(15,16)(19,20)(23,26)(24,27)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=fg,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=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307649, #72307862, #72307935
maximal normal subgroup of	#72307649, #72307862, #72307935

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7472710);