#7627300

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
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×#3494	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534496);