#7527580

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2132 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257604, #72257637, #72263860, …
maximal normal subgroup of	#72257604, #72257637, #72263860, …
maximal quotient of	#72307861

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7434776);