#7682041

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#2404	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,5,7)(3,8,4,6)(9,10,13,15)(11,12,14,16)(17,18,23,20)(19,21,25,24)(22,28,30,26)(27,31,32,29);(1,3,5,4)(2,6,7,8)(9,11,13,14)(10,12,15,16)(17,19,23,25)(18,24,20,21)(22,27,30,32)(26,31,28,29);(1,2)(3,6)(4,8)(5,7)(9,10)(11,12)(13,15)(14,16)(18,20)(19,25)(22,30)(29,31);(1,4,5,3)(2,6,7,8)(9,12)(10,14)(11,15)(13,16)(17,20,23,18)(19,21,25,24)(22,28,30,26)(27,29,32,31);(17,21)(18,25)(19,20)(22,29)(23,24)(26,32)(27,28)(30,31);(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249479, #72249502, #72249532
maximal normal subgroup of	#72249479, #72249502, #72249532

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7589237);