#7626284

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
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×#3487	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8394 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)(5,6,9,7)(8,11,12,10)(14,16)(15,20)(18,26)(25,28);(6,7)(10,11)(13,14,19,16)(15,22,20,17)(18,21,26,24)(23,28,27,25);(1,3,4,2)(5,7,9,6)(8,11,12,10)(13,15)(14,17)(16,22)(18,23)(19,20)(21,25)(24,28)(26,27);(5,8)(6,10)(7,11)(9,12)(13,16,19,14)(15,22,20,17)(18,24,26,21)(23,28,27,25);(13,17,19,22)(14,20,16,15)(18,25,26,28)(21,27,24,23);(13,18)(14,21)(15,23)(16,24)(17,25)(19,26)(20,27)(22,28)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256632, #72256694, #72256697, …
maximal normal subgroup of	#72256632, #72256694, #72256697, …
maximal quotient of	#72263795

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533480);