#7627597

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3499	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 44 points:

(1,2,3,4)(5,6,9,7)(8,11,12,10)(14,19)(20,30)(21,31)(22,32)(33,41)(34,42)(35,36)(43,44);(6,7)(10,11)(14,19)(15,23)(21,31)(22,32)(24,37)(25,38)(35,36)(39,40);(5,7,9,6)(8,11,12,10)(13,14,18,19)(15,20,23,30)(16,21,27,31)(17,22,29,32)(24,33,37,41)(25,34,38,42)(26,35,28,36)(39,43,40,44);(5,8)(6,10)(7,11)(9,12)(13,15,18,23)(14,20,19,30)(16,24,27,37)(17,25,29,38)(21,33,31,41)(22,34,32,42)(26,39,28,40)(35,43,36,44);(13,16,18,27)(14,21,19,31)(15,24,23,37)(17,28,29,26)(20,33,30,41)(22,36,32,35)(25,40,38,39)(34,44,42,43);(13,17,18,29)(14,22,19,32)(15,25,23,38)(16,26,27,28)(20,34,30,42)(21,35,31,36)(24,39,37,40)(33,43,41,44)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534793);