#7627359

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×#3540	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7176 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)(7,10)(8,11)(9,12)(13,14)(15,22)(16,21)(17,20)(18,23)(19,24)(25,27)(26,28);(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,25)(17,26)(19,23)(20,27)(21,28)(22,24);(1,2,4,3)(5,7)(6,9)(8,12)(10,11)(13,16,19,17)(14,20,24,21)(15,25,23,26)(18,27,22,28);(13,17,19,16)(14,21,24,20)(15,26,23,25)(18,28,22,27);(1,3,4,2)(5,7)(6,10)(8,12)(9,11)(13,15)(14,22)(16,25)(17,26)(18,24)(19,23)(20,28)(21,27);(5,6)(7,9)(8,11)(10,12)(13,18,19,22)(14,23,24,15)(16,27,17,28)(20,26,21,25)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=g,e^2=i,f^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, 7534555);